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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 433 (page 1069)

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

Short Answer

Expert verified

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

Step by step solution

01

Step 1. Given data

The given Logarithmic Equation is $\log_{2} x + \log_{2} (x - 2) = 3$

02

Step 2. Apply the product properties of logarithm

Use product property of logarithm $\log_{a} (m 路 n) = \log_{a} (m) + \log_{a} (n)$

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

03

Step 3. Write the equation in exponential form

Change the logarithm equation into an exponential equation

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

so $x = - 2, 4$

The logarithm of a negative number is not possible

so $x = 4$

04

Step 4. Verification

Substitute $4$ for $x$ in the equation

$\log_{2} x + \log_{2} (x - 2) = 3 \log_{2} 4 + \log_{2} (4 - 2) \overset{?}{=} 3 \log_{2} 4 + \log_{2} 2 \overset{?}{=} 3 \log_{2} (4 路 2) \overset{?}{=} 3 \log_{2} 2^{3} \overset{?}{=} 3 3 = 3$

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