Question

In the following exercises, find the exact value of each logarithm without using a calculator.

${\log }_8\left(1\right)$

Short answer

The value is $0$.

Step-by-step solution

Step 1. Form a logarithmic equation and write its equivalent exponential form 

Consider the expression ${\log }_8\left(1\right)$.

Setting the expression equal to $x$we get,

${\log }_8\left(1\right)=x$

And its equivalent exponential form is given as $8^x=1$.

Step 2. Compare the exponents 

The number $1$can be equivalently written as $8^0$. So we get

$$\begin{gathered} 8^x=1 \\ {8}^x=8^0 \end{gathered}$$

Now, as the base is the same, so on comparing the exponents we get

$x=0$

Thus the value of${\log }_8\left(1\right)$is $0$.