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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. 435 (page 1065)

In the following exercises, solve for x.
$\log_{4} (x - 2) + \log_{4} (x + 5) = \log_{4} 8$

Short Answer

Expert verified

The value of x for the equation $\log_{4} (x - 2) + \log_{4} (x + 5) = \log_{4} 8$ is $3$

Step by step solution

01

Step 1. Given data

The given Logarithmic Equation is $\log_{4} (x - 2) + \log_{4} (x + 5) = \log_{4} 8$

02

Step 2. Apply the product properties of logarithm

Use product property of the logarithm

$\log_{4} (x - 2) + \log_{4} (x + 5) = \log_{4} 8 \log_{4} (x - 2) (x + 5) = \log_{4} 8 \log_{4} (x^{2} + 3 x - 10) = \log_{4} 8$

03

Step 3. Apply one to one property of logarithms

Use one to one property of logarithms

$\log_{4} (x^{2} + 3 x - 10) = \log_{4} 8 x^{2} + 3 x - 10 = 8 x^{2} + 3 x - 18 = 0 (x - 3) (x + 6) = 0$

so $x = 3, - 6$

the logarithm of a negative number is not possible

so $x = 3$

04

Step 4. verification

Substitute $3$ for x in the equation

$\log_{4} (x - 2) + \log_{4} (x + 5) = \log_{4} 8 \log_{4} (3 - 2) + \log_{4} (3 + 5) \overset{?}{=} \log_{4} 8 \log_{4} 1 + \log_{4} (8) \overset{?}{=} \log_{4} 8 0 + \log_{4} (8) \overset{?}{=} \log_{4} 8 \log_{4} (8) = \log_{4} 8$

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