Question

Find the exact value of each logarithm without using a calculator:

$$\begin{gathered} a){\log }_{12}\left(144\right) \\ b){\log }_4\left(2\right) \\ c){\log }_2\left(\frac{1}{32}\right) \end{gathered}$$

Short answer

Part (a) The value of the logarithm is $2$.

Part (b) The value of the logarithm is $\frac{1}{2}$.

Part (c) The value of the logarithm is$-5$.

Step-by-step solution

Step 1. Given information.

The given logarithms are

$$\begin{gathered} a){\log }_{12}\left(144\right) \\ b){\log }_4\left(2\right) \\ c){\log }_2\left(\frac{1}{32}\right) \end{gathered}$$

Step 2. Solving part (a).

Consider the given expression,

${\log }_{12}\left(144\right)=x$

Now, we will convert the given expression to exponential form and simplify.

$$\begin{gathered} {\log }_{12}\left(144\right)=x \\ {\left(12\right)}^x=144 \\ {\left(12\right)}^x={\left(12\right)}^2 \\ x=2 \end{gathered}$$

Hence, the value of the logarithm is$2$.

Step 3. Solving part (b).

Consider the given expression,

${\log }_4\left(2\right)=x$

Now, we will convert the given expression to exponential form and simplify.

$$\begin{gathered} {\log }_4\left(2\right)=x \\ {4}^x=2 \\ {\left(2^2\right)}^x=2 \\ {2}^{2x}=2^1 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

Hence, the value of the logarithm is $\frac{1}{2}$.

Step 4. Solving part (c).

Consider the given expression,

${\log }_2\left(\frac{1}{32}\right)=x$

Now, we will convert the given expression to exponential form and simplify.

$$\begin{gathered} {\log }_2\left(\frac{1}{32}\right)=x \\ {2}^x=\frac{1}{32} \\ {2}^x=\frac{1}{2^5} \\ {2}^x=2^{-5} \\ x=-5 \end{gathered}$$

Hence, the value of the logarithm is$-5$.