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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. 206 (page 1035)

In the following exercises, solve each logarithmic equation.
$\log_{2} (x^{2} - 4) = 5$

Short Answer

Expert verified

The solutions are $x = 6$ and

$x = - 6$ .

Step by step solution

01

Step 1. Rewrite in exponential form and simplify

Consider the logarithmic equation $\log_{2} (x^{2} - 4) = 5$ .

Its equivalent exponential form is given as $2^{5} = x^{2} - 4$

It can be simplified to

$32 = x^{2} - 4$

02

Step 2. Solve the quadratic equation for x

Set the left hand side equal to zero, by subtracting $32$ from both sides.

$\begin{array}{rcl} 32 - 32 & = & x^{2} - 4 - 32 \\ 0 & = & x^{2} - 36 \end{array}$

Factor the right hand side and solve the equation.

$\begin{array}{rcl} 0 & = & x^{2} - 6^{2} \\ 0 & = & (x - 6) (x + 6) \end{array}$

So the solutions are

$\begin{array}{rcl} x - 6 & = & 0 \\ x & = & 6 \end{array}$

and

$\begin{array}{rcl} x + 6 & = & 0 \\ x & = & - 6 \end{array}$

03

Step 3. Check for the first solution

Substitute $6$ for $x$ in the original equation, we get

$\begin{array}{rcl} \log_{2} (x^{2} - 4) & = & 5 \\ \log_{2} (6^{2} - 4) & = & 5 \\ \log_{2} (36 - 4) & = & 5 \\ \log_{2} (32) & = & 5 \\ 2^{5} & = & 32 \\ 32 & = & 32 \end{array}$

So we get a true statement, thus the found solution is correct.

04

Step 4. Check for the second solution

Substitute $- 6$ for $x$ in the original equation, we get

$\begin{array}{rcl} \log_{2} (x^{2} - 4) & = & 5 \\ \log_{2} ({(- 6)}^{2} - 4) & = & 5 \\ \log_{2} (36 - 4) & = & 5 \\ \log_{2} (32) & = & 5 \\ 2^{5} & = & 32 \\ 32 & = & 32 \end{array}$

So we get a true statement, thus the found solution is correct.

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Most popular questions from this chapter

In the following exercises, determine whether or not the given functions are inverses.

$f (x) = x - 9 a n d g (x) = x + 9$

In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.
$\{(5, 3), (4, 2), (3, 1), (2, 0)\}$ .

Find the inverse of the function , $f (x) = \sqrt[4]{6 x - 7}$

Verify that the functions are inverse functions.

$f (x) = 2 x + 6$

$g (x) = \frac{x - 6}{2}$

In the following exercises, find the inverse of each function.

$f (x) = - 2 x + 5$

See all solutions

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