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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 297 (page 1069)

In the following exercise, solve for x.
$\log_{3} x + \log_{3} (x + 6) = 3$

Short Answer

Expert verified

The solution of the given expression is $x = 3$

Step by step solution

01

Step 1. Given information.

The given expression is $\log_{3} x + \log_{3} (x + 6) = 3$

02

Step 2. Use logarithmic properties

Using product rule, we get:

$\log_{3} x + \log_{3} (x + 6) = 3 \log_{3} (x) (x + 6) = 3 \log_{3} (x^{2} + 6 x) = 3$

Writing it in exponential form, we get:

$\log_{3} (x^{2} + 6 x) = 3 x + 6 x = 3^{3} x + 6 x - 27 = 0$

03

Step 3. Solve for x.

Solving the equation, we get:

$x^{2} + 6 x - 27 = 0 x^{2} + 9 x - 3 x - 27 = 0 x (x + 9) - 3 (x + 9) = 0 (x - 3) (x + 9) = 0$

$x = 3$ or

$x = - 9$

We eliminate $x = - 9$ as the logarithmic function doesn't take any negative values.

04

Step 4. Check the value

Substituting $x = 1$ in the given expression, we get:

$\log_{3} x + \log_{3} (x + 6) = 3 \log_{3} 3 + \log_{3} (3 + 6) = 3 1 + \log_{3} 9 = 3 1 + 2 = 3 3 = 3$

This is true.

Thus, $x = 3$ is the solution of the given function.

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