Question

In the following exercises, solve for x.

${\log }_{\frac{1}{2}}\left(\frac{1}{16}\right)=x$

Short answer

The solution is $x=4$.

Step-by-step solution

Step 1. Convert into exponential form`

Consider the logarithmic equation ${\log }_{\frac{1}{2}}\left(\frac{1}{16}\right)=x$.

Its equivalent exponential form is given as ${\left(\frac{1}{2}\right)}^x=\frac{1}{16}$.

Step 2. Compare the exponents

The number $\frac{1}{16}$can be written as ${\left(\frac{1}{2}\right)}^4$. So the exponential equation is given as

$$\begin{gathered} {\left(\frac{1}{2}\right)}^x=\frac{1}{16} \\ {\left(\frac{1}{2}\right)}^x={\left(\frac{1}{2}\right)}^4 \end{gathered}$$

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

$x=4$

Step 3. Check the solution

Substitute $4$for $x$in the original equation.

$\begin{array}{rcl}{\log }_{\frac{1}{2}}\left(\frac{1}{16}\right)& =& x\\ {\log }_{\frac{1}{2}}\left(\frac{1}{16}\right)& =& 4\\ {\left(\frac{1}{2}\right)}^4& =& \frac{1}{16}\\ \frac{1}{16}& =& \frac{1}{16}\end{array}$

As the statement is true, so the found solution is correct.