Convert Between Logarithmic and Exponential

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base $b$, exponent $x$, and output $y$. Then we write $x=log_b(y)$.

EXAMPLE 2

Converting from Exponential Form to Logarithmic Form

Write the following exponential equations in logarithmic form.

1. $2^3=8$
2. $5^2=25$
3. $10^{−4}=\frac{1}{10,000}$

Solution

First, identify the values of $b,y$, and $x$. Then, write the equation in the form $x=log_b(y)$.

1. $2^3=8$
Here, $b=2, x=3$, and $y=8$. Therefore, the equation $2^3=8$ is equivalent to $log_2(8)=3$.
2. $5^2=25$
   
Here, $b=5, x=2$, and $y=25$. Therefore, the equation $5^2=25$ is equivalent to $log_5(25)=2$.
3. $10^{−4}=\frac{1}{10,000}$
   
Here, $b=10, x=−4$, and $y= \frac{1}{10,000}$. Therefore, the equation $10^{−4}= \frac{1}{10,000}$ is equivalent to $log_{10}(\frac{1}{10,000})=−4$.

TRY IT #2

Write the following exponential equations in logarithmic form.

ⓐ $3^2=9$

ⓑ $5^3=125$

ⓒ $2{−1}=\frac{1}{2}$