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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. 305 (page 1054)

In the following exercises, solve for $x$
$\log (5 x + 1) = \log (x + 3) + \log 2$

Short Answer

Expert verified

By solving the equation $\log (5 x + 1) = \log (x + 3) + \log 2$ , the value of $x$ is $\frac{5}{3}$

Step by step solution

01

Step 1. Given

An expression $\log (5 x + 1) = \log (x + 3) + \log 2$

To solve the expression for $x$

02

Step 2. Use the Product property

By Product Property of logarithm,

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

$\log (5 x + 1) = \log (x + 3) . (2) \log (5 x + 1) = \log (2 x + 6)$

03

Step 3. Use One to One property

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

$5 x + 1 = 2 x + 6 5 x - 2 x = 6 - 1 3 x = 5 x = \frac{5}{3}$

The value of $x$ is $\frac{5}{3}$

04

Step 4. Check the solution

Substitute $x = \frac{5}{3}$ in the original equation and solve.

$\log (5 (\frac{5}{3}) + 1) = \log ((\frac{5}{3}) + 3) + \log 2 \log (\frac{25}{3} + 1) = \log (\frac{14}{3}) . (2) \log \frac{28}{3} = \log \frac{28}{3}$

Hence the equation has been checked.

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