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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. 293 (page 1054)

In the following exercise, solve for x.
$\log_{3} (x^{2} + 3) = \log_{3} 4 x$

Short Answer

Expert verified

The solution of the given expression is $x = 1 x = 3$

Step by step solution

01

Step 1. Given information.

The given expression is $\log_{3} (x^{2} + 3) = \log_{3} 4 x$

02

Step 2. Use logarithmic properties

Using one to one property, we get:

$\log_{3} (x^{2} + 3) = \log_{3} 4 x (x^{2} + 3) = 4 x x^{2} - 4 x + 3 = 0$

03

Step 3. Solve for x.

Solving the equation, we get:

$x^{2} - 4 x + 3 = 0 x^{2} - 3 x - x + 3 = 0 x (x - 3) - (x - 3) = 0 [t a k i n g o u t x a s c o m m o n] (x - 1) (x - 3) = 0 [t a k i n g o u t (x - 3) a s c o m m o n]$

So,

$x = 1$ or

$x = 3$

04

Step 4. Check the value

Substituting $x = 3$ in the given expression, we get:

$\log_{3} (x^{2} + 3) = \log_{3} 4 x \log_{3} (3^{2} + 3) = \log_{3} 4 路 3 \log_{3} (9 + 3) = \log_{3} 12 \log_{3} 12 = = \log_{3} 12$

This is true

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

$\log_{3} (x^{2} + 3) = \log_{3} 4 x \log_{3} (1^{2} + 3) = \log_{3} 4 路 1 \log_{3} (4) = \log_{3} 4$

This is true.

Thus, $x = 3 x = 1$ is the solution of the given function.

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