Logarithmic Functions: Detailed Explanation and 100 Examples

Introduction to Logarithmic Functions

A logarithmic function is the inverse of an exponential function. If $a^y=x$, then the logarithmic form is written as ${\log }_a(x)=y$, which reads as “the logarithm of $x$ to the base $a$ equals $y$.” Logarithms help us deal with very large or very small numbers by expressing them as exponents.

Key properties of logarithms include:

• ${\log }_a(1)=0$ for any base $a$, because $a^0=1$.
• ${\log }_a(a)=1$, because $a^1=a$.
• ${\log }_a(xy)={\log }_a(x)+{\log }_a(y)$.
• ${\log }_a\left(\frac{x}{y}\right)={\log }_a(x)–{\log }_a(y)$.
• ${\log }_a(x^k)=k{\log }_a(x)$.
• Change of base formula: ${\log }_a(x)=\frac{{\log }_b(x)}{{\log }_b(a)}$, where $b$ is any positive base.

Example 1: Evaluate ${\log }_2(8)$

Solution:

Step 1: Rewrite $8$ as $2^3$.
${\log }_2(8)={\log }_2(2^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_2(8)=3$.

---

Example 2: Solve for $x$ in the equation ${\log }_3(x)=4$

Solution:

Step 1: Rewrite the equation in exponential form.
$3^4=x$

Step 2: Evaluate $3^4$.
$=81$

Thus, $x=81$.

---

Example 3: Simplify ${\log }_5(25)$

Solution:

Step 1: Rewrite $25$ as $5^2$.
${\log }_5(25)={\log }_5(5^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_5(25)=2$.

---

Example 4: Use the change of base formula to find ${\log }_2(10)$

Solution:

Step 1: Use the change of base formula.
${\log }_2(10)=\frac{{\log }_{10}(10)}{{\log }_{10}(2)}$

Step 2: Evaluate using a calculator (using ${\log }_{10}$).
$=\frac{1}{0.3010}$

$=3.3219$

Thus, ${\log }_2(10)\approx 3.322$.

---

Example 5: Simplify ${\log }_3(9)+{\log }_3(3)$

Solution:

Step 1: Rewrite $9$ as $3^2$ and use ${\log }_3(9)=2$.
${\log }_3(9)+{\log }_3(3)=2+1$

Step 2: Add the values.
$=3$

Thus, ${\log }_3(9)+{\log }_3(3)=3$.

---

Example 6: Solve $2{\log }_5(x)=4$

Solution:

Step 1: Divide both sides by 2.
${\log }_5(x)=2$

Step 2: Rewrite in exponential form.
$5^2=x$

Step 3: Evaluate $5^2$.
$x=25$

Thus, $x=25$.

---

Example 7: Simplify ${\log }_4(64)$

Solution:

Step 1: Rewrite $64$ as $4^3$.
${\log }_4(64)={\log }_4(4^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_4(64)=3$.

---

Example 8: Solve ${\log }_7(x)+{\log }_7(2)=1$

Solution:

Step 1: Combine the logarithms.
${\log }_7(2x)=1$

Step 2: Rewrite in exponential form.
$7^1=2x$

Step 3: Solve for $x$.
$7=2x$
$x=\frac{7}{2}$

Thus, $x=3.5$.

---

Example 9: Solve ${\log }_3(x^2)=5$

Solution:

Step 1: Use the property ${\log }_a(x^k)=k{\log }_a(x)$.
$2{\log }_3(x)=5$

Step 2: Divide by 2.
${\log }_3(x)=\frac{5}{2}$

Step 3: Rewrite in exponential form.
$x=3^{\frac{5}{2}}$

Step 4: Evaluate $3^{\frac{5}{2}}$.
$x=\sqrt{3^5}=\sqrt{243}\approx 15.588$

Thus, $x\approx 15.588$.

---

Example 10: Simplify ${\log }_2(16)–{\log }_2(4)$

Solution:

Step 1: Use the property ${\log }_a\left(\frac{x}{y}\right)={\log }_a(x)–{\log }_a(y)$.
${\log }_2\left(\frac{16}{4}\right)={\log }_2(4)$

Step 2: Rewrite $4$ as $2^2$.
${\log }_2(4)=2$

Thus, ${\log }_2(16)–{\log }_2(4)=2$.

---

Example 11: Solve for $x$ in the equation ${\log }_6(36)=x$

Solution:

Step 1: Rewrite $36$ as $6^2$.
${\log }_6(36)={\log }_6(6^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$x=2$

Thus, ${\log }_6(36)=2$.

---

Example 12: Simplify ${\log }_5(25)+{\log }_5(5)$

Solution:

Step 1: Use the property ${\log }_a(a^k)=k$.
${\log }_5(25)=2$
${\log }_5(5)=1$

Step 2: Add the values.
$=2+1$

Thus, ${\log }_5(25)+{\log }_5(5)=3$.

---

Example 13: Use the change of base formula to find ${\log }_3(7)$

Solution:

Step 1: Use the change of base formula.
${\log }_3(7)=\frac{{\log }_{10}(7)}{{\log }_{10}(3)}$

Step 2: Evaluate using a calculator (using ${\log }_{10}$).
$=\frac{0.8451}{0.4771}$

$=1.7712$

Thus, ${\log }_3(7)\approx 1.771$.

---

Example 14: Solve ${\log }_4(x)=3$

Solution:

Step 1: Rewrite the equation in exponential form.
$4^3=x$

Step 2: Evaluate $4^3$.
$=64$

Thus, $x=64$.

---

Example 15: Simplify ${\log }_7(49)$

Solution:

Step 1: Rewrite $49$ as $7^2$.
${\log }_7(49)={\log }_7(7^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_7(49)=2$.

---

Example 16: Solve $2{\log }_3(x)=6$

Solution:

Step 1: Divide both sides by 2.
${\log }_3(x)=3$

Step 2: Rewrite in exponential form.
$x=3^3$

Step 3: Evaluate $3^3$.
$x=27$

Thus, $x=27$.

---

Example 17: Simplify ${\log }_5(125)$

Solution:

Step 1: Rewrite $125$ as $5^3$.
${\log }_5(125)={\log }_5(5^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_5(125)=3$.

---

Example 18:

Solve ${\log }_4(x)+{\log }_4(2)=3$

Solution:

Step 1: Combine the logarithms using the property ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_4(2x)=3$

Step 2: Rewrite in exponential form.
$4^3=2x$

Step 3: Solve for $x$.
$64=2x$
$x=\frac{64}{2}$
$x=32$

Thus, $x=32$.

---

Example 19:

Simplify ${\log }_8(64)$

Solution:

Step 1: Use the change of base formula:
${\log }_8(64)=\frac{{\log }_{10}(64)}{{\log }_{10}(8)}$

Step 2: Evaluate using a calculator (using ${\log }_{10}$).
$=\frac{1.806}{0.903}$

$=2$

Thus, ${\log }_8(64)=2$.

---

Example 20:

Solve $3{\log }_2(x)=9$

Solution:

Step 1: Divide both sides by 3.
${\log }_2(x)=3$

Step 2: Rewrite in exponential form.
$2^3=x$

Step 3: Solve for $x$.
$x=8$

Thus, $x=8$.

---

Example 21:

Simplify ${\log }_4(16)–{\log }_4(2)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_4\left(\frac{16}{2}\right)={\log }_4(8)$

Step 2: Rewrite $8$ as $4^{3/2}$.
${\log }_4(8)=\frac{3}{2}$

Thus, ${\log }_4(16)–{\log }_4(2)=\frac{3}{2}$.

---

Example 22:

Solve for $x$ in ${\log }_6(x^2)=4$

Solution:

Step 1: Use the property ${\log }_a(x^k)=k{\log }_a(x)$.
$2{\log }_6(x)=4$

Step 2: Divide by 2.
${\log }_6(x)=2$

Step 3: Rewrite in exponential form.
$6^2=x$

Step 4: Evaluate $6^2$.
$x=36$

Thus, $x=36$.

---

Example 23:

Simplify ${\log }_2(32)$

Solution:

Step 1: Rewrite $32$ as $2^5$.
${\log }_2(32)={\log }_2(2^5)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=5$

Thus, ${\log }_2(32)=5$.

---

Example 24:

Solve ${\log }_5(x)–{\log }_5(4)=1$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_5\left(\frac{x}{4}\right)=1$

Step 2: Rewrite in exponential form.
$5^1=\frac{x}{4}$

Step 3: Solve for $x$.
$5=\frac{x}{4}$
$x=5\times 4$
$x=20$

Thus, $x=20$.

---

Example 25:

Simplify ${\log }_3(27)+{\log }_3(9)$

Solution:

Step 1: Use the property ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_3(27\times 9)={\log }_3(243)$

Step 2: Rewrite $243$ as $3^5$.
${\log }_3(243)=5$

Thus, ${\log }_3(27)+{\log }_3(9)=5$.

---

Example 26:

Solve for $x$: $2{\log }_4(x)=6$

Solution:

Step 1: Divide both sides by 2.
${\log }_4(x)=3$

Step 2: Rewrite in exponential form.
$4^3=x$

Step 3: Solve for $x$.
$x=64$

Thus, $x=64$.

---

Example 27:

Simplify ${\log }_6(36)–{\log }_6(6)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_6\left(\frac{36}{6}\right)={\log }_6(6)$

Step 2: Use ${\log }_6(6)=1$.
$=1$

Thus, ${\log }_6(36)–{\log }_6(6)=1$.

---

Example 28:

Solve ${\log }_9(x)+{\log }_9(3)=2$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_9(3x)=2$

Step 2: Rewrite in exponential form.
$9^2=3x$

Step 3: Solve for $x$.
$81=3x$
$x=\frac{81}{3}$
$x=27$

Thus, $x=27$.

---

Example 29:

Simplify ${\log }_7(343)$

Solution:

Step 1: Rewrite $343$ as $7^3$.
${\log }_7(343)={\log }_7(7^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_7(343)=3$.

---

Example 30:

Solve for $x$ in the equation ${\log }_2(x+4)=5$

Solution:

Step 1: Rewrite in exponential form.
$2^5=x+4$

Step 2: Evaluate $2^5$.
$32=x+4$

Step 3: Solve for $x$.
$x=32–4$
$x=28$

Thus, $x=28$.

---

Example 31:

Simplify ${\log }_3(81)$

Solution:

Step 1: Rewrite $81$ as $3^4$.
${\log }_3(81)={\log }_3(3^4)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=4$

Thus, ${\log }_3(81)=4$.

---

Example 32:

Solve ${\log }_5(x)=-1$

Solution:

Step 1: Rewrite in exponential form.
$5^{-1}=x$

Step 2: Solve for $x$.
$x=\frac{1}{5}$

Thus, $x=\frac{1}{5}$.

---

Example 33:

Simplify ${\log }_2(16)+{\log }_2(4)$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_2(16\times 4)={\log }_2(64)$

Step 2: Rewrite $64$ as $2^6$.
${\log }_2(64)=6$

Thus, ${\log }_2(16)+{\log }_2(4)=6$.

---

Example 34:

Solve $3{\log }_7(x)=9$

Solution:

Step 1: Divide both sides by 3.
${\log }_7(x)=3$

Step 2: Rewrite in exponential form.
$7^3=x$

Step 3: Evaluate $7^3$.
$x=343$

Thus, $x=343$.

---

Example 35:

Simplify ${\log }_4(256)$

Solution:

Step 1: Rewrite $256$ as $4^4$.
${\log }_4(256)={\log }_4(4^4)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=4$

Thus, ${\log }_4(256)=4$.

---

Example 36:

Solve ${\log }_3(x+2)=4$

Solution:

Step 1: Rewrite in exponential form.
$3^4=x+2$

Step 2: Evaluate $3^4$.
$81=x+2$

Step 3: Solve for $x$.
$x=81–2$
$x=79$

Thus, $x=79$.

---

Example 37:

Simplify ${\log }_2(32)–{\log }_2(8)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_2\left(\frac{32}{8}\right)={\log }_2(4)$

Step 2: Rewrite $4$ as $2^2$.
${\log }_2(4)=2$

Thus, ${\log }_2(32)–{\log }_2(8)=2$.

---

Example 38:

Solve $2{\log }_3(x)=8$

Solution:

Step 1: Divide both sides by 2.
${\log }_3(x)=4$

Step 2: Rewrite in exponential form.
$3^4=x$

Step 3: Evaluate $3^4$.
$x=81$

Thus, $x=81$.

---

Example 39:

Simplify ${\log }_4(64)$

Solution:

Step 1: Use the change of base formula:
${\log }_4(64)=\frac{{\log }_{10}(64)}{{\log }_{10}(4)}$

Step 2: Use a calculator to evaluate.
${\log }_{10}(64)=1.806$
${\log }_{10}(4)=0.602$
$\frac{1.806}{0.602}=3$

Thus, ${\log }_4(64)=3$.

---

Example 40:

Solve for $x$ in ${\log }_5(x–1)=2$

Solution:

Step 1: Rewrite in exponential form.
$5^2=x–1$

Step 2: Solve for $x$.
$25=x–1$
$x=25+1$
$x=26$

Thus, $x=26$.

---

Example 41:

Simplify ${\log }_3(81)–{\log }_3(9)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_3\left(\frac{81}{9}\right)={\log }_3(9)$

Step 2: Rewrite $9$ as $3^2$.
${\log }_3(9)=2$

Thus, ${\log }_3(81)–{\log }_3(9)=2$.

---

Example 42:

Solve ${\log }_4(x)+{\log }_4(2)=3$

Solution:

Step 1: Use the property ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_4(2x)=3$

Step 2: Rewrite in exponential form.
$4^3=2x$

Step 3: Solve for $x$.
$64=2x$
$x=\frac{64}{2}$
$x=32$

Thus, $x=32$.

---

Example 43:

Simplify ${\log }_7(343)$

Solution:

Step 1: Rewrite $343$ as $7^3$.
${\log }_7(343)={\log }_7(7^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_7(343)=3$.

---

Example 44:

Solve $3{\log }_6(x)=6$

Solution:

Step 1: Divide both sides by 3.
${\log }_6(x)=2$

Step 2: Rewrite in exponential form.
$6^2=x$

Step 3: Solve for $x$.
$x=36$

Thus, $x=36$.

---

Example 45:

Simplify ${\log }_{1}0(100)$

Solution:

Step 1: Rewrite $100$ as ${10}^2$.
${\log }_{1}0(100)={\log }_{1}0({10}^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_{1}0(100)=2$.

---

Example 46:

Solve ${\log }_5(x+2)=3$

Solution:

Step 1: Rewrite in exponential form.
$5^3=x+2$

Step 2: Evaluate $5^3$.
$125=x+2$

Step 3: Solve for $x$.
$x=125–2$
$x=123$

Thus, $x=123$.

---

Example 47:

Simplify ${\log }_2(16)$

Solution:

Step 1: Rewrite $16$ as $2^4$.
${\log }_2(16)={\log }_2(2^4)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=4$

Thus, ${\log }_2(16)=4$.

---

Example 48:

Solve ${\log }_8(x)=1$

Solution:

Step 1: Rewrite in exponential form.
$8^1=x$

Step 2: Solve for $x$.
$x=8$

Thus, $x=8$.

---

Example 49:

Simplify ${\log }_9(81)$

Solution:

Step 1: Rewrite $81$ as $9^2$.
${\log }_9(81)={\log }_9(9^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_9(81)=2$.

---

Example 50:

Solve for $x$ in ${\log }_3(x–1)=4$

Solution:

Step 1: Rewrite in exponential form.
$3^4=x–1$

Step 2: Evaluate $3^4$.
$81=x–1$

Step 3: Solve for $x$.
$x=81+1$
$x=82$

Thus, $x=82$.

---

Example 51:

Simplify ${\log }_2(32)$

Solution:

Step 1: Rewrite $32$ as $2^5$.
${\log }_2(32)={\log }_2(2^5)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=5$

Thus, ${\log }_2(32)=5$.

---

Example 52:

Solve ${\log }_4(x)+{\log }_4(4)=2$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_4(4x)=2$

Step 2: Rewrite in exponential form.
$4^2=4x$

Step 3: Solve for $x$.
$16=4x$
$x=\frac{16}{4}$
$x=4$

Thus, $x=4$.

---

Example 53:

Simplify ${\log }_3(9)+{\log }_3(27)$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_3(9\times 27)={\log }_3(243)$

Step 2: Rewrite $243$ as $3^5$.
${\log }_3(243)=5$

Thus, ${\log }_3(9)+{\log }_3(27)=5$.

---

Example 54:

Solve for $x$ in $4{\log }_2(x)=8$

Solution:

Step 1: Divide both sides by 4.
${\log }_2(x)=2$

Step 2: Rewrite in exponential form.
$2^2=x$

Step 3: Solve for $x$.
$x=4$

Thus, $x=4$.

---

Example 55:

Simplify ${\log }_7(49)$

Solution:

Step 1: Rewrite $49$ as $7^2$.
${\log }_7(49)={\log }_7(7^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_7(49)=2$.

---

Example 56:

Solve for $x$: ${\log }_6(x)=3$

Solution:

Step 1: Rewrite in exponential form.
$6^3=x$

Step 2: Solve for $x$.
$x=216$

Thus, $x=216$.

---

Example 57:

Simplify ${\log }_5(25)–{\log }_5(5)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_5\left(\frac{25}{5}\right)={\log }_5(5)$

Step 2: Use ${\log }_5(5)=1$.
$=1$

Thus, ${\log }_5(25)–{\log }_5(5)=1$.

---

Example 58:

Solve for $x$ in ${\log }_4(x^2)=3$

Solution:

Step 1: Use the property ${\log }_a(x^k)=k{\log }_a(x)$.
$2{\log }_4(x)=3$

Step 2: Divide by 2.
${\log }_4(x)=\frac{3}{2}$

Step 3: Rewrite in exponential form.
$x=4^{\frac{3}{2}}$

Step 4: Evaluate $4^{\frac{3}{2}}=\sqrt{4^3}=\sqrt{64}=8$

Thus, $x=8$.

---

Example 59:

Simplify ${\log }_3(81)–{\log }_3(9)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_3\left(\frac{81}{9}\right)={\log }_3(9)$

Step 2: Rewrite $9$ as $3^2$.
${\log }_3(9)=2$

Thus, ${\log }_3(81)–{\log }_3(9)=2$.

---

Example 60:

Solve $5{\log }_5(x)=10$

Solution:

Step 1: Divide both sides by 5.
${\log }_5(x)=2$

Step 2: Rewrite in exponential form.
$5^2=x$

Step 3: Solve for $x$.
$x=25$

Thus, $x=25$.

---

Example 61:

Simplify ${\log }_6(36)$

Solution:

Step 1: Rewrite $36$ as $6^2$.
${\log }_6(36)={\log }_6(6^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_6(36)=2$.

---

Example 62:

Solve for $x$: ${\log }_7(x+2)=3$

Solution:

Step 1: Rewrite in exponential form.
$7^3=x+2$

Step 2: Evaluate $7^3$.
$343=x+2$

Step 3: Solve for $x$.
$x=343–2$
$x=341$

Thus, $x=341$.

---

Example 63:

Simplify ${\log }_5(125)–{\log }_5(25)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_5\left(\frac{125}{25}\right)={\log }_5(5)$

Step 2: Use ${\log }_5(5)=1$.
$=1$

Thus, ${\log }_5(125)–{\log }_5(25)=1$.

---

Example 64:

Solve $4{\log }_3(x)=8$

Solution:

Step 1: Divide both sides by 4.
${\log }_3(x)=2$

Step 2: Rewrite in exponential form.
$3^2=x$

Step 3: Solve for $x$.
$x=9$

Thus, $x=9$.

---

Example 65:

Simplify ${\log }_{1}0(1000)$

Solution:

Step 1: Rewrite $1000$ as ${10}^3$.
${\log }_{1}0(1000)={\log }_{1}0({10}^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_{1}0(1000)=3$.

---

Example 66:

Solve ${\log }_9(x–2)=2$

Solution:

Step 1: Rewrite in exponential form.
$9^2=x–2$

Step 2: Evaluate $9^2$.
$81=x–2$

Step 3: Solve for $x$.
$x=81+2$
$x=83$

Thus, $x=83$.

---

Example 67:

Simplify ${\log }_2(8)+{\log }_2(4)$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_2(8\times 4)={\log }_2(32)$

Step 2: Rewrite $32$ as $2^5$.
${\log }_2(32)=5$

Thus, ${\log }_2(8)+{\log }_2(4)=5$.

---

Example 68:

Solve for $x$: ${\log }_5(x^2)=6$

Solution:

Step 1: Use the property ${\log }_a(x^k)=k{\log }_a(x)$.
$2{\log }_5(x)=6$

Step 2: Divide by 2.
${\log }_5(x)=3$

Step 3: Rewrite in exponential form.
$5^3=x$

Step 4: Solve for $x$.
$x=125$

Thus, $x=125$.

---

Example 69:

Simplify ${\log }_7(49)+{\log }_7(7)$

Solution:

Step 1: Use the property ${\log }_a(a^k)=k$.
${\log }_7(49)=2$
${\log }_7(7)=1$

Step 2: Add the values.
$2+1=3$

Thus, ${\log }_7(49)+{\log }_7(7)=3$.

---

Example 70:

Solve $5{\log }_4(x)=10$

Solution:

Step 1: Divide both sides by 5.
${\log }_4(x)=2$

Step 2: Rewrite in exponential form.
$4^2=x$

Step 3: Solve for $x$.
$x=16$

Thus, $x=16$.

---

Example 71:

Simplify ${\log }_6(36)–{\log }_6(6)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_6\left(\frac{36}{6}\right)={\log }_6(6)$

Step 2: Use ${\log }_6(6)=1$.
$=1$

Thus, ${\log }_6(36)–{\log }_6(6)=1$.

---

Example 72:

Solve ${\log }_3(x+1)=4$

Solution:

Step 1: Rewrite in exponential form.
$3^4=x+1$

Step 2: Evaluate $3^4$.
$81=x+1$

Step 3: Solve for $x$.
$x=81–1$
$x=80$

Thus, $x=80$.

---

Example 73:

Simplify ${\log }_2(64)$

Solution:

Step 1: Rewrite $64$ as $2^6$.
${\log }_2(64)={\log }_2(2^6)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=6$

Thus, ${\log }_2(64)=6$.

---

Example 74:

Solve $3{\log }_4(x)=6$

Solution:

Step 1: Divide both sides by 3.
${\log }_4(x)=2$

Step 2: Rewrite in exponential form.
$4^2=x$

Step 3: Solve for $x$.
$x=16$

Thus, $x=16$.

---

Example 75:

Simplify ${\log }_8(64)$

Solution:

Step 1: Use the change of base formula:
${\log }_8(64)=\frac{{\log }_{10}(64)}{{\log }_{10}(8)}$

Step 2: Evaluate using a calculator.
$=\frac{1.806}{0.903}=2$

Thus, ${\log }_8(64)=2$.

---

Example 76:

Solve for $x$ in ${\log }_5(x–4)=2$

Solution:

Step 1: Rewrite in exponential form.
$5^2=x–4$

Step 2: Evaluate $5^2$.
$25=x–4$

Step 3: Solve for $x$.
$x=25+4$
$x=29$

Thus, $x=29$.

---

Example 77:

Simplify ${\log }_2(32)+{\log }_2(16)$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_2(32\times 16)={\log }_2(512)$

Step 2: Rewrite $512$ as $2^9$.
${\log }_2(512)=9$

Thus, ${\log }_2(32)+{\log }_2(16)=9$.

---

Example 78:

Solve ${\log }_4(x)+{\log }_4(2)=3$

Solution:

Step 1: Combine the logarithms.
${\log }_4(2x)=3$

Step 2: Rewrite in exponential form.
$4^3=2x$

Step 3: Solve for $x$.
$64=2x$
$x=\frac{64}{2}$
$x=32$

Thus, $x=32$.

---

Example 79:

Simplify ${\log }_7(343)$

Solution:

Step 1: Rewrite $343$ as $7^3$.
${\log }_7(343)={\log }_7(7^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_7(343)=3$.

---

Example 80:

Solve ${\log }_2(x+1)=5$

Solution:

Step 1: Rewrite in exponential form.
$2^5=x+1$

Step 2: Evaluate $2^5$.
$32=x+1$

Step 3: Solve for $x$.
$x=32–1$
$x=31$

Thus, $x=31$.

---

Example 81:

Simplify ${\log }_{1}0(100)$

Solution:

Step 1: Rewrite $100$ as ${10}^2$.
${\log }_{1}0(100)={\log }_{1}0({10}^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_{1}0(100)=2$.

---

Example 82:

Solve $2{\log }_5(x)=6$

Solution:

Step 1: Divide both sides by 2.
${\log }_5(x)=3$

Step 2: Rewrite in exponential form.
$5^3=x$

Step 3: Solve for $x$.
$x=125$

Thus, $x=125$.

---

Example 83:

Simplify ${\log }_3(81)$

Solution:

Step 1: Rewrite $81$ as $3^4$.
${\log }_3(81)={\log }_3(3^4)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=4$

Thus, ${\log }_3(81)=4$.

---

Example 84:

Solve for $x$: ${\log }_5(x–3)=2$

Solution:

Step 1: Rewrite in exponential form.
$5^2=x–3$

Step 2: Evaluate $5^2$.
$25=x–3$

Step 3: Solve for $x$.
$x=25+3$
$x=28$

Thus, $x=28$.

---

Example 85:

Simplify ${\log }_7(49)$

Solution:

Step 1: Rewrite $49$ as $7^2$.
${\log }_7(49)={\log }_7(7^2)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=2$

Thus, ${\log }_7(49)=2$.

---

Example 86:

Solve ${\log }_8(x)+{\log }_8(4)=2$

Solution:

Step 1: Combine the logarithms.
${\log }_8(4x)=2$

Step 2: Rewrite in exponential form.
$8^2=4x$

Step 3: Solve for $x$.
$64=4x$
$x=\frac{64}{4}$
$x=16$

Thus, $x=16$.

---

Example 87:

Simplify ${\log }_2(64)–{\log }_2(8)$

Solution:

Step 1: Use the property ${\log }_a(x)–{\log }_a(y)={\log }_a\left(\frac{x}{y}\right)$.
${\log }_2\left(\frac{64}{8}\right)={\log }_2(8)$

Step 2: Rewrite $8$ as $2^3$.
${\log }_2(8)=3$

Thus, ${\log }_2(64)–{\log }_2(8)=3$.

---

Example 88:

Solve ${\log }_5(x)=1$

Solution:

Step 1: Rewrite in exponential form.
$5^1=x$

Step 2: Solve for $x$.
$x=5$

Thus, $x=5$.

---

Example 89:

Simplify ${\log }_{1}0(1000)$

Solution:

Step 1: Rewrite $1000$ as ${10}^3$.
${\log }_{1}0(1000)={\log }_{1}0({10}^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_{1}0(1000)=3$.

---

Example 90:

Solve for $x$: $2{\log }_4(x)=4$

Solution:

Step 1: Divide both sides by 2.
${\log }_4(x)=2$

Step 2: Rewrite in exponential form.
$4^2=x$

Step 3: Solve for $x$.
$x=16$

Thus, $x=16$.

---

Example 91:

Simplify ${\log }_6(216)$

Solution:

Step 1: Rewrite $216$ as $6^3$.
${\log }_6(216)={\log }_6(6^3)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=3$

Thus, ${\log }_6(216)=3$.

---

Example 92:

Solve ${\log }_3(x+1)=2$

Solution:

Step 1: Rewrite in exponential form.
$3^2=x+1$

Step 2: Solve for $x$.
$9=x+1$
$x=9–1$
$x=8$

Thus, $x=8$.

---

Example 93:

Simplify ${\log }_2(64)$

Solution:

Step 1: Rewrite $64$ as $2^6$.
${\log }_2(64)={\log }_2(2^6)$

Step 2: Use the property ${\log }_a(a^k)=k$.
$=6$

Thus, ${\log }_2(64)=6$.

---

Example 94:

Solve $3{\log }_5(x)=9$

Solution:

Step 1: Divide both sides by 3.
${\log }_5(x)=3$

Step 2: Rewrite in exponential form.
$5^3=x$

Step 3: Solve for $x$.
$x=125$

Thus, $x=125$.

---

Example 95:

Simplify ${\log }_{1}0(10)$

Solution:

Step 1: Use the property ${\log }_a(a)=1$.
${\log }_{1}0(10)=1$

Thus, ${\log }_{1}0(10)=1$.

---

Example 96:

Solve for $x$: ${\log }_2(x–3)=5$

Solution:

Step 1: Rewrite in exponential form.
$2^5=x–3$

Step 2: Solve for $x$.
$32=x–3$
$x=32+3$
$x=35$

Thus, $x=35$.

---

Example 97:

Simplify ${\log }_3(27)+{\log }_3(9)$

Solution:

Step 1: Use the property ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_3(27\times 9)={\log }_3(243)$

Step 2: Rewrite $243$ as $3^5$.
${\log }_3(243)=5$

Thus, ${\log }_3(27)+{\log }_3(9)=5$.

---

Example 98:

Solve ${\log }_4(x^2)=4$

Solution:

Step 1: Use the property ${\log }_a(x^k)=k{\log }_a(x)$.
$2{\log }_4(x)=4$

Step 2: Divide by 2.
${\log }_4(x)=2$

Step 3: Rewrite in exponential form.
$4^2=x$

Step 4: Solve for $x$.
$x=16$

Thus, $x=16$.

---

Example 99:

Simplify ${\log }_2(8)+{\log }_2(4)$

Solution:

Step 1: Combine the logarithms using ${\log }_a(x)+{\log }_a(y)={\log }_a(xy)$.
${\log }_2(8\times 4)={\log }_2(32)$

Step 2: Rewrite $32$ as $2^5$.
${\log }_2(32)=5$

Thus, ${\log }_2(8)+{\log }_2(4)=5$.

---

Example 100:

Solve ${\log }_6(x+4)=3$

Solution:

Step 1: Rewrite in exponential form.
$6^3=x+4$

Step 2: Evaluate $6^3$.
$216=x+4$

Step 3: Solve for $x$.
$x=216–4$
$x=212$

Thus, $x=212$.