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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. 300 (page 1054)

In the following exercises, solve for $x$
$\log (x + 4) - \log (5 x + 12) = - \log x$

Short Answer

Expert verified

By solving the equation $\log (x + 4) - \log (5 x + 12) = - \log x$ , the value of $x$ is $4$

Step by step solution

01

Step 1. Given

An expression $\log (x + 4) - \log (5 x + 12) = - \log x$

To solve the expression for $x$

02

Step 2. Use the Quotient property

By Quotient Property of logarithm,

$\log_{a} M - \log_{a} N = \log_{a} \frac{M}{N}$

$\log (x + 4) - \log (5 x + 12) = - \log x \log (\frac{x + 4}{5 x + 12}) = \log x^{- 1} \log (\frac{x + 4}{5 x + 12}) = \log \frac{1}{x}$

03

Step 3. Use One to One property

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

So, $\frac{x + 4}{5 x + 12} = \frac{1}{x}$

04

Step 4. Solve the equation

$\frac{x + 4}{5 x + 12} = \frac{1}{x}$

$x^{2} - x - 12 = 0 x^{2} - 4 x + 3 x - 12 = 0 (x - 4) (x + 3) = 0 x = - 3, 4$

Logarithm for a negative number does not exist.

So, $x = 4$

05

Step 5. Check the solution

Substitute $x = 4$ in the original equation and solve.

$\log (4 + 4) - \log (5 (4) + 12) = - \log (4) \log 8 - \log 32 = \log 4^{- 1} \log \frac{8}{32} = \log \frac{1}{4} \log \frac{1}{4} = \log \frac{1}{4}$

Hence the solution is checked.

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