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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. 304 (page 1054)

In the following exercises, solve for $x$
$\log_{3} (2 x - 1) = \log_{3} (x + 3) + \log_{3} 3$

Short Answer

Expert verified

By solving the equation $\log_{3} (2 x - 1) = \log_{3} (x + 3) + \log_{3} 3$ , the value of $x$ is $- 10$

Step by step solution

01

Step 1. Given

An expression $\log_{3} (2 x - 1) = \log_{3} (x + 3) + \log_{3} 3$

To solve the expression for $x$

02

Step 2. Use the Product property

By Product Property of logarithm,

$\log_{a} M + \log_{a} N = \log_{a} M N$

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

03

Step 3. Use One to One property

If $\log_{a} M = \log_{a} N$ , then $M = N$

$2 x - 1 = 3 x + 9 3 x - 2 x = - 9 - 1 x = - 10$

04

Step 5. Check the solution

Substitute $x = - 10$ in the original equation and solve.

$\log_{3} (2 (- 10) - 1) = \log_{3} ((- 10) + 3) + \log_{3} 3 \log_{3} (- 21) = \log_{3} (- 7) + \log_{3} 3 \log_{3} (- 21) = \log_{3} (- 21)$

Hence the solution is checked.

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