Converting logarithms into exponentials

We can also convert logarithmic expressions into exponential ones using the same exact definition of logarithms:

\log_a(b) = c \quad \text{is equivalent to} \quad a^c = b

Convert \log_2(8)=3 into an exponential expression

• \log_a(b) = c is equivalent to a^c = b
• a=2, b=8, and c=3
• 2^3=8

Convert \log_5(25)=2 into an exponential expression

• \log_a(b) = c is equivalent to a^c = b
• 5^2=25

Convert \log_3(27)=3 into an exponential expression

• \log_a(b) = c is equivalent to a^c = b
• 3^3=27

Write as an exponential expression: \log_{10}(1000)=x

• \log_a(b) = c is equivalent to a^c = b
• 10^x=1000

Write as an exponential expression: \ln(7)=x

• \ln(7)=\log_e(7)
• \log_a(b) = c is equivalent to a^c = b
• e^x=7

Write as an exponential expression: \log_4(x)=2

• \log_a(b) = c is equivalent to a^c = b
• 4^2=x

Write as an exponential expression: \log_2(64)=3x

• \log_a(b) = c is equivalent to a^c = b
• 2^{3x}=64

Write as an exponential expression: \log_5(125)=2(2x-1)

• \log_a(b) = c is equivalent to a^c = b
• 5^{2(2x-1)}=125
• 5^{4x-2}=125

Write as an exponential expression: \log_y(27)=3

• \log_a(b) = c is equivalent to a^c = b
• y^3=27

Write as an exponential expression: \log_3(9)=x+1

• \log_a(b) = c is equivalent to a^c = b
• 3^{x+1}=9

Write as an exponential expression: \log_{0.5}(0.125)=x-2

• \log_a(b) = c is equivalent to a^c = b
• (0.5)^{x-2}=0.125

Write as an exponential expression: \log_{x}(y)=z

• \log_a(b) = c is equivalent to a^c = b
• x^z=y

Write as an exponential expression: \log_{3x}(\frac{y-4}2)=z+1

• \log_a(b) = c is equivalent to a^c = b
• (3x)^{z+1}=\frac{y-4}2

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