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

Step 1. Given data

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

Step 2. Apply the product properties of logarithm

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

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

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 $x = 4$

Step 4. Verification

Substitute $4$ for $x$ in the equation

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

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

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the product properties of logarithm

Step 3. Write the equation in exponential form

Step 4. Verification

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In the following exercises, solve for x.log2x+log2x-2=3

Short Answer

Step by step solution

Step 1. Given data

Step 2. Apply the product properties of logarithm

Step 3. Write the equation in exponential form

Step 4. Verification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Statistics

Calculus

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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