Exponents and Logarithms

Exponents and logarithms are essential mathematical concepts that help simplify complex calculations involving powers and roots. They are widely used in algebra, calculus, and many areas of science and engineering. In this blog, we will explore the fundamentals of exponents and logarithms, along with 100 detailed examples that demonstrate their applications. Each example will be explained step by step, ensuring clarity and understanding.

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1. Exponents

1.1 What Are Exponents?

An exponent refers to the number of times a number, called the base, is multiplied by itself. For example, in the expression $2^3$, 2 is the base, and 3 is the exponent. This means $2$ is multiplied by itself three times:

$2^3=2\times 2\times 2=8$

Exponents are a shorthand way to express repeated multiplication.

1.2 Laws of Exponents

Here are the key laws of exponents:

1. Product Rule: $a^m\times a^n=a^{m+n}$
   • Example: $2^3\times 2^4=2^{3+4}=2^7=128$
2. Quotient Rule: $\frac{a^m}{a^n}=a^{m-n}$
   • Example: $\frac{5^6}{5^3}=5^{6-3}=5^3=125$
3. Power of a Power Rule: $(a^m{)}^n=a^{m\times n}$
   • Example: $(2^4{)}^2=2^{4\times 2}=2^8=256$
4. Power of a Product Rule: $(ab{)}^n=a^n\times b^n$
   • Example: $(3\times 4{)}^2=3^2\times 4^2=9\times 16=144$
5. Power of a Quotient Rule: ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$
   • Example: ${\left(\frac{2}{3}\right)}^2=\frac{2^2}{3^2}=\frac{4}{9}$
6. Zero Exponent Rule: $a^0=1$, provided $a\ne 0$
   • Example: $5^0=1$
7. Negative Exponent Rule: $a^{-n}=\frac{1}{a^n}$
   • Example: $4^{-2}=\frac{1}{4^2}=\frac{1}{16}$

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2. Logarithms

2.1 What Are Logarithms?

A logarithm is the inverse operation of exponentiation. It tells us the power to which a base must be raised to get a specific number. If $a^x=b$, then ${\log }_{a}b=x$.

For example, if $2^3=8$, then ${\log }_{2}8=3$.

2.2 Laws of Logarithms

Here are the fundamental laws of logarithms:

1. Product Rule: ${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$
   • Example: ${\log }_2(4\times 8)={\log }_{2}4+{\log }_{2}8=2+3=5$
2. Quotient Rule: ${\log }_b\left(\frac{x}{y}\right)={\log }_{b}x–{\log }_{b}y$
   • Example: ${\log }_3\left(\frac{9}{3}\right)={\log }_{3}9–{\log }_{3}3=2–1=1$
3. Power Rule: ${\log }_b(x^n)=n{\log }_{b}x$
   • Example: ${\log }_2(8^2)=2{\log }_{2}8=2\times 3=6$
4. Change of Base Rule: ${\log }_{b}x=\frac{{\log }_{k}x}{{\log }_{k}b}$
   • Example: ${\log }_{5}25=\frac{{\log }_{10}25}{{\log }_{10}5}=\frac{1.3979}{0.6989}=2$
5. Logarithm of 1: ${\log }_{b}1=0$
   • Example: ${\log }_{4}1=0$
6. Logarithm of Base: ${\log }_{b}b=1$
   • Example: ${\log }_{7}7=1$

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3. 100 Examples of Exponents and Logarithms

Let’s explore 100 detailed examples with step-by-step solutions to solidify your understanding of exponents and logarithms.

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Exponents Examples

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Example 1: Simplify $2^4\times 2^3$.

Solution:
$2^4\times 2^3=2^{4+3}$
$=2^7$
$=128$

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Example 2: Simplify $\frac{5^6}{5^2}$.

Solution:
$\frac{5^6}{5^2}=5^{6-2}$
$=5^4$
$=625$

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Example 3: Simplify $(3^2{)}^3$.

Solution:
$(3^2{)}^3=3^{2\times 3}$
$=3^6$
$=729$

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Example 4: Simplify $(5\times 6{)}^2$.

Solution:
$(5\times 6{)}^2=5^2\times 6^2$
$=25\times 36$
$=900$

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Example 5: Simplify ${\left(\frac{9}{3}\right)}^2$.

Solution:
${\left(\frac{9}{3}\right)}^2=\frac{9^2}{3^2}$
$=\frac{81}{9}$
$=9$

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Example 6: Simplify $4^0$.

Solution:
$4^0=1$

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Example 7: Simplify $6^{-2}$.

Solution:
$6^{-2}=\frac{1}{6^2}$
$=\frac{1}{36}$

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Example 8: Simplify ${\left(\frac{2}{3}\right)}^4$.

Solution:
${\left(\frac{2}{3}\right)}^4=\frac{2^4}{3^4}$
$=\frac{16}{81}$

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Example 9: Simplify $(7^3{)}^2$.

Solution:
$(7^3{)}^2=7^{3\times 2}$
$=7^6$
$=117649$

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Example 10: Simplify $\frac{3^5}{3^3}$.

Solution:
$\frac{3^5}{3^3}=3^{5-3}$
$=3^2$
$=9$

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Logarithms Examples

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Example 11: Solve ${\log }_{2}16$.

Solution:
${\log }_{2}16=x\Rightarrow 2^x=16$
$2^4=16$
$x=4$

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Example 12: Solve ${\log }_{5}125$.

Solution:
${\log }_{5}125=x\Rightarrow 5^x=125$
$5^3=125$
$x=3$

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Example 13: Solve ${\log }_{3}27$.

Solution:
${\log }_{3}27=x\Rightarrow 3^x=27$
$3^3=27$
$x=3$

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Example 14: Solve ${\log }_{4}64$.

Solution:
${\log }_{4}64=x\Rightarrow 4^x=64$
$4^3=64$
$x=3$

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Example 15: Solve ${\log }_{7}49$.

Solution:
${\log }_{7}49=x\Rightarrow 7^x=49$
$7^2=49$
$x=2$

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Example 16: Simplify ${\log }_3(9\times 27)$.

Solution:
${\log }_3(9\times 27)={\log }_{3}9+{\log }_{3}27$
$=2+3$
$=5$

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Example 17: Simplify ${\log }_5\left(\frac{25}{5}\right)$.

Solution:
${\log }_5\left(\frac{25}{5}\right)={\log }_{5}25–{\log }_{5}5$
$=2–1$
$=1$

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Example 18: Simplify ${\log }_2(8^2)$.

Solution:
${\log }_2(8^2)=2{\log }_{2}8$
$=2\times 3$
$=6$

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Example 19: Solve ${\log }_{4}1$.

Solution:
${\log }_{4}1=0$

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Example 20: Solve ${\log }_{5}5$.

Solution:
${\log }_{5}5=1$


Exponents Examples

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Example 21: Simplify $\frac{6^7}{6^3}$.

Solution:
$\frac{6^7}{6^3}=6^{7-3}$
$=6^4$
$=1296$

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Example 22: Simplify $(4^2{)}^3$.

Solution:
$(4^2{)}^3=4^{2\times 3}$
$=4^6$
$=4096$

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Example 23: Simplify $2^5\times 2^3$.

Solution:
$2^5\times 2^3=2^{5+3}$
$=2^8$
$=256$

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Example 24: Simplify $\frac{{10}^4}{{10}^2}$.

Solution:
$\frac{{10}^4}{{10}^2}={10}^{4-2}$
$={10}^2$
$=100$

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Example 25: Simplify $(3^2\times 3^3)$.

Solution:
$3^2\times 3^3=3^{2+3}$
$=3^5$
$=243$

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Example 26: Simplify $7^{-2}\times 7^5$.

Solution:
$7^{-2}\times 7^5=7^{(-2+5)}$
$=7^3$
$=343$

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Example 27: Simplify $(5^3{)}^2\times 5^{-4}$.

Solution:
$(5^3{)}^2=5^6$
$5^6\times 5^{-4}=5^{6-4}$
$=5^2$
$=25$

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Example 28: Simplify $\frac{2^6}{2^2}\times 2^{-3}$.

Solution:
$\frac{2^6}{2^2}=2^{6-2}=2^4$
$2^4\times 2^{-3}=2^{4-3}=2^1$
$=2$

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Example 29: Simplify $3^4÷9^2$.

Solution:
$3^4÷(3^2{)}^2=\frac{3^4}{3^4}$
$=1$

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Example 30: Simplify $(2^3\times 2^4)÷2^5$.

Solution:
$2^3\times 2^4=2^{3+4}=2^7$
$\frac{2^7}{2^5}=2^{7-5}=2^2$
$=4$

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Example 31: Simplify $\frac{8^3}{4^3}$.

Solution:
$\frac{8^3}{4^3}=\frac{(2^3{)}^3}{(2^2{)}^3}$
$=\frac{2^9}{2^6}$
$=2^{9-6}$
$=2^3$
$=8$

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Example 32: Simplify $(7^4{)}^{\frac{1}{2}}$.

Solution:
$(7^4{)}^{\frac{1}{2}}=7^{4\times \frac{1}{2}}$
$=7^2$
$=49$

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Example 33: Simplify $\frac{5^5}{5^3}\times 5^2$.

Solution:
$\frac{5^5}{5^3}=5^{5-3}=5^2$
$5^2\times 5^2=5^{2+2}$
$=5^4$
$=625$

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Example 34: Simplify $\frac{9^3}{3^4}$.

Solution:
$\frac{9^3}{3^4}=\frac{(3^2{)}^3}{3^4}$
$=\frac{3^6}{3^4}$
$=3^{6-4}=3^2$
$=9$

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Example 35: Simplify $4^3\times 2^{-6}$.

Solution:
$4^3=(2^2{)}^3=2^6$
$2^6\times 2^{-6}=2^{6-6}$
$=2^0$
$=1$

Logarithms Examples

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Example 36: Solve ${\log }_{2}32$.

Solution:
${\log }_{2}32=x\Rightarrow 2^x=32$
$2^5=32$
$x=5$

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Example 37: Solve ${\log }_{5}25$.

Solution:
${\log }_{5}25=x\Rightarrow 5^x=25$
$5^2=25$
$x=2$

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Example 38: Simplify ${\log }_3(27\times 9)$.

Solution:
${\log }_3(27\times 9)={\log }_{3}27+{\log }_{3}9$
$=3+2$
$=5$

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Example 39: Solve ${\log }_7\left(\frac{49}{7}\right)$.

Solution:
${\log }_7\left(\frac{49}{7}\right)={\log }_{7}49–{\log }_{7}7$
$=2–1$
$=1$

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Example 40: Solve ${\log }_{2}64$.

Solution:
${\log }_{2}64=x\Rightarrow 2^x=64$
$2^6=64$
$x=6$

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Example 41: Simplify ${\log }_3(9^2)$.

Solution:
${\log }_3(9^2)=2{\log }_{3}9$
$=2\times 2$
$=4$

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Example 42: Solve ${\log }_5\left(\frac{125}{25}\right)$.

Solution:
${\log }_5\left(\frac{125}{25}\right)={\log }_{5}125–{\log }_{5}25$
$=3–2$
$=1$

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Example 43: Solve ${\log }_{2}1$.

Solution:
${\log }_{2}1=0$

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Example 44: Solve ${\log }_{4}16$.

Solution:
${\log }_{4}16=x\Rightarrow 4^x=16$
$4^2=16$
$x=2$

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Example 45: Solve ${\log }_{9}81$.

Solution:
${\log }_{9}81=x\Rightarrow 9^x=81$
$9^2=81$
$x=2$

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Example 46: Simplify ${\log }_2(4\times 8)$.

Solution:
${\log }_2(4\times 8)={\log }_{2}4+{\log }_{2}8$
$=2+3$
$=5$

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Example 47: Simplify ${\log }_6\left(\frac{36}{6}\right)$.

Solution:
${\log }_6\left(\frac{36}{6}\right)={\log }_{6}36–{\log }_{6}6$
$=2–1$
$=1$

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Example 48: Solve ${\log }_{7}49$.

Solution:
${\log }_{7}49=x\Rightarrow 7^x=49$
$7^2=49$
$x=2$

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Example 49: Solve ${\log }_{3}81$.

Solution:
${\log }_{3}81=x\Rightarrow 3^x=81$
$3^4=81$
$x=4$

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Example 50: Solve ${\log }_{1}01000$.

Solution:
${\log }_{1}01000=x\Rightarrow {10}^x=1000$
${10}^3=1000$
$x=3$

Exponents Examples

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Example 51: Simplify $4^5÷2^6$.

Solution:
$\frac{4^5}{2^6}=\frac{(2^2{)}^5}{2^6}$
$=\frac{2^{10}}{2^6}$
$=2^{10-6}$
$=2^4$
$=16$

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Example 52: Simplify $(3^5{)}^{\frac{1}{5}}$.

Solution:
$(3^5{)}^{\frac{1}{5}}=3^{5\times \frac{1}{5}}$
$=3^1$
$=3$

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Example 53: Simplify $8^3÷4^2$.

Solution:
$\frac{8^3}{4^2}=\frac{(2^3{)}^3}{(2^2{)}^2}$
$=\frac{2^9}{2^4}$
$=2^{9-4}$
$=2^5$
$=32$

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Example 54: Simplify $5^{-3}\times 5^4$.

Solution:
$5^{-3}\times 5^4=5^{-3+4}$
$=5^1$
$=5$

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Example 55: Simplify ${\left(\frac{3}{4}\right)}^2$.

Solution:
${\left(\frac{3}{4}\right)}^2=\frac{3^2}{4^2}$
$=\frac{9}{16}$

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Example 56: Simplify $(6^2)\times (6^{-4})$.

Solution:
$6^2\times 6^{-4}=6^{2+(-4)}$
$=6^{-2}$
$=\frac{1}{6^2}$
$=\frac{1}{36}$

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Example 57: Simplify $2^7÷2^5$.

Solution:
$\frac{2^7}{2^5}=2^{7-5}$
$=2^2$
$=4$

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Example 58: Simplify $(4^3{)}^2$.

Solution:
$(4^3{)}^2=4^{3\times 2}$
$=4^6$
$=4096$

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Example 59: Simplify $\frac{9^3}{3^4}$.

Solution:
$\frac{9^3}{3^4}=\frac{(3^2{)}^3}{3^4}$
$=\frac{3^6}{3^4}$
$=3^{6-4}$
$=3^2$
$=9$

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Example 60: Simplify $(5^4{)}^{\frac{1}{2}}$.

Solution:
$(5^4{)}^{\frac{1}{2}}=5^{4\times \frac{1}{2}}$
$=5^2$
$=25$

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Logarithms Examples

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Example 61: Solve ${\log }_{2}8$.

Solution:
${\log }_{2}8=x\Rightarrow 2^x=8$
$2^3=8$
$x=3$

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Example 62: Solve ${\log }_{3}9$.

Solution:
${\log }_{3}9=x\Rightarrow 3^x=9$
$3^2=9$
$x=2$

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Example 63: Solve ${\log }_{5}125$.

Solution:
${\log }_{5}125=x\Rightarrow 5^x=125$
$5^3=125$
$x=3$

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Example 64: Simplify ${\log }_2(4\times 8)$.

Solution:
${\log }_2(4\times 8)={\log }_{2}4+{\log }_{2}8$
$=2+3$
$=5$

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Example 65: Simplify ${\log }_3(9÷3)$.

Solution:
${\log }_3(9÷3)={\log }_{3}9–{\log }_{3}3$
$=2–1$
$=1$

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Example 66: Solve ${\log }_{1}01000$.

Solution:
${\log }_{1}01000=x\Rightarrow {10}^x=1000$
${10}^3=1000$
$x=3$

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Example 67: Simplify ${\log }_5(25\times 5)$.

Solution:
${\log }_5(25\times 5)={\log }_{5}25+{\log }_{5}5$
$=2+1$
$=3$

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Example 68: Solve ${\log }_{2}16$.

Solution:
${\log }_{2}16=x\Rightarrow 2^x=16$
$2^4=16$
$x=4$

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Example 69: Solve ${\log }_{3}81$.

Solution:
${\log }_{3}81=x\Rightarrow 3^x=81$
$3^4=81$
$x=4$

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Example 70: Solve ${\log }_{4}64$.

Solution:
${\log }_{4}64=x\Rightarrow 4^x=64$
$4^3=64$
$x=3$

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Example 71: Simplify ${\log }_2(8\times 16)$.

Solution:
${\log }_2(8\times 16)={\log }_{2}8+{\log }_{2}16$
$=3+4$
$=7$

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Example 72: Simplify ${\log }_3(27÷9)$.

Solution:
${\log }_3(27÷9)={\log }_{3}27–{\log }_{3}9$
$=3–2$
$=1$

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Example 73: Solve ${\log }_{7}49$.

Solution:
${\log }_{7}49=x\Rightarrow 7^x=49$
$7^2=49$
$x=2$

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Example 74: Solve ${\log }_{9}81$.

Solution:
${\log }_{9}81=x\Rightarrow 9^x=81$
$9^2=81$
$x=2$

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Example 75: Solve ${\log }_{5}625$.

Solution:
${\log }_{5}625=x\Rightarrow 5^x=625$
$5^4=625$
$x=4$

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Example 76: Simplify ${\log }_3(9\times 27)$.

Solution:
${\log }_3(9\times 27)={\log }_{3}9+{\log }_{3}27$
$=2+3$
$=5$

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Example 77: Solve ${\log }_2(16\times 32)$.

Solution:
${\log }_2(16\times 32)={\log }_{2}16+{\log }_{2}32$
$=4+5$
$=9$

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Example 78: Simplify ${\log }_4(64÷16)$.

Solution:
${\log }_4(64÷16)={\log }_{4}64–{\log }_{4}16$
$=3–2$
$=1$

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Example 79: Solve ${\log }_{8}64$.

Solution:
${\log }_{8}64=x\Rightarrow 8^x=64$
$8^2=64$
$x=2$

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Example 80: Solve ${\log }_{6}36$.

Solution:
${\log }_{6}36=x\Rightarrow 6^x=36$
$6^2=36$
$x=2$

Example 81: Solve ${\log }_{2}128$.

Solution:
${\log }_{2}128=x\Rightarrow 2^x=128$
$2^7=128$
$x=7$

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Example 82: Simplify ${\log }_3(81\times 9)$.

Solution:
${\log }_3(81\times 9)={\log }_{3}81+{\log }_{3}9$
$=4+2$
$=6$

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Example 83: Solve ${\log }_{1}0100$.

Solution:
${\log }_{1}0100=x\Rightarrow {10}^x=100$
${10}^2=100$
$x=2$

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Example 84: Simplify ${\log }_5\left(\frac{25}{5}\right)$.

Solution:
${\log }_5\left(\frac{25}{5}\right)={\log }_{5}25–{\log }_{5}5$
$=2–1$
$=1$

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Example 85: Solve ${\log }_{7}343$.

Solution:
${\log }_{7}343=x\Rightarrow 7^x=343$
$7^3=343$
$x=3$

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Example 86: Simplify ${\log }_6(216\times 36)$.

Solution:
${\log }_6(216\times 36)={\log }_{6}216+{\log }_{6}36$
$=3+2$
$=5$

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Example 87: Solve ${\log }_{9}729$.

Solution:
${\log }_{9}729=x\Rightarrow 9^x=729$
$9^3=729$
$x=3$

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Example 88: Solve ${\log }_3(81÷9)$.

Solution:
${\log }_3(81÷9)={\log }_{3}81–{\log }_{3}9$
$=4–2$
$=2$

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Example 89: Solve ${\log }_{2}256$.

Solution:
${\log }_{2}256=x\Rightarrow 2^x=256$
$2^8=256$
$x=8$

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Example 90: Simplify ${\log }_4\left(\frac{64}{16}\right)$.

Solution:
${\log }_4\left(\frac{64}{16}\right)={\log }_{4}64–{\log }_{4}16$
$=3–2$
$=1$

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Example 91: Solve ${\log }_{2}32$.

Solution:
${\log }_{2}32=x\Rightarrow 2^x=32$
$2^5=32$
$x=5$

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Example 92: Solve ${\log }_{4}256$.

Solution:
${\log }_{4}256=x\Rightarrow 4^x=256$
$4^4=256$
$x=4$

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Example 93: Simplify ${\log }_5(25\times 125)$.

Solution:
${\log }_5(25\times 125)={\log }_{5}25+{\log }_{5}125$
$=2+3$
$=5$

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Example 94: Solve ${\log }_{8}512$.

Solution:
${\log }_{8}512=x\Rightarrow 8^x=512$
$8^3=512$
$x=3$

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Example 95: Simplify ${\log }_3(81\times 27)$.

Solution:
${\log }_3(81\times 27)={\log }_{3}81+{\log }_{3}27$
$=4+3$
$=7$

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Example 96: Solve ${\log }_4(64\times 16)$.

Solution:
${\log }_4(64\times 16)={\log }_{4}64+{\log }_{4}16$
$=3+2$
$=5$

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Example 97: Solve ${\log }_{9}243$.

Solution:
${\log }_{9}243=x\Rightarrow 9^x=243$
$9^{\frac{5}{2}}=243$
$x=\frac{5}{2}$

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Example 98: Simplify ${\log }_6(216÷36)$.

Solution:
${\log }_6(216÷36)={\log }_{6}216–{\log }_{6}36$
$=3–2$
$=1$

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Example 99: Solve ${\log }_{2}64$.

Solution:
${\log }_{2}64=x\Rightarrow 2^x=64$
$2^6=64$
$x=6$

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Example 100: Simplify ${\log }_3(81\times 9)$.

Solution:
${\log }_3(81\times 9)={\log }_{3}81+{\log }_{3}9$
$=4+2$
$=6$