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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 10: Q. 399 (page 1064)

In the following exercises, solve for x.
$\log_{\frac{1}{2}} (\frac{1}{16}) = x$

Short Answer

Expert verified

The solution is $x = 4$ .

Step by step solution

01

Step 1. Convert into exponential form`

Consider the logarithmic equation $\log_{\frac{1}{2}} (\frac{1}{16}) = x$ .

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

02

Step 2. Compare the exponents

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

${(\frac{1}{2})}^{x} = \frac{1}{16} {(\frac{1}{2})}^{x} = {(\frac{1}{2})}^{4}$

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

$x = 4$

03

Step 3. Check the solution

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

$\begin{array}{rcl} \log_{\frac{1}{2}} (\frac{1}{16}) & = & x \\ \log_{\frac{1}{2}} (\frac{1}{16}) & = & 4 \\ {(\frac{1}{2})}^{4} & = & \frac{1}{16} \\ \frac{1}{16} & = & \frac{1}{16} \end{array}$

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

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