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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. 207 (page 1035)

In the following exercises, solve each logarithmic equation.
$\log_{3} (x^{2} + 2) = 3$

Short Answer

Expert verified

The solutions are $x = 5$ and

$x = - 5$ .

Step by step solution

01

Step 1. Rewrite in exponential form and simplify

Consider the logarithmic equation $\log_{3} (x^{2} + 2) = 3$ .

Its equivalent exponential form is given as $3^{3} = x^{2} + 2$

It can be simplified to

$27 = x^{2} + 2$

02

Step 2. Solve the quadratic equation for x

Set the left hand side equal to zero, by subtracting $27$ from both sides.

$\begin{array}{rcl} 27 - 27 & = & x^{2} + 2 - 27 \\ 0 & = & x^{2} - 25 \end{array}$

Factor the right hand side and solve the equation.

$0 = x^{2} - 5^{2} 0 = (x - 5) (x + 5)$

So the solutions are

$\begin{array}{rcl} x - 5 & = & 0 \\ x & = & 5 \end{array}$

and

$\begin{array}{rcl} x + 5 & = & 0 \\ x & = & - 5 \end{array}$

03

Step 3. Check for the first solution

Substitute $5$ for $x$ in the original equation, we get

$\begin{array}{rcl} \log_{3} (x^{2} + 2) & = & 3 \\ \log_{3} (5^{2} + 2) & = & 3 \\ \log_{3} (25 + 2) & = & 3 \\ \log_{3} (27) & = & 3 \\ 3^{3} & = & 27 \\ 27 & = & 27 \end{array}$

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

04

Step 4. Check for the second solution

Substitute $- 5$ for $x$ in the original equation, we get

$\begin{array}{rcl} \log_{3} (x^{2} + 2) & = & 3 \\ \log_{3} ({(- 5)}^{2} + 2) & = & 3 \\ \log_{3} (25 + 2) & = & 3 \\ \log_{3} (27) & = & 3 \\ 3^{3} & = & 27 \\ 27 & = & 27 \end{array}$

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, graph, on the same coordinate system, the inverse of the one-to-one function shown.

In the following exercises, match the graphs to one of the following functions:

$鈸� 2^{x} 鈸� 2^{x + 1} 鈸� 2^{x 鈭� 1} 鈸� 2^{x} + 2 鈸� 2^{x} 鈭� 2 鈸� 3^{x}$

Convert from exponential to logarithmic form:

$17^{x} = \sqrt[5]{17}$ .

In the following exercises, find

a. $f 鈭� g (x)$

b. $g 鈭� f (x)$

c. $(f . g) (x)$

On the same coordinate system, graph:

$f (x) = 2^{x}$

$g (x) = 2^{x - 1}$

See all solutions

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