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Chapter 3: Problem 38

Find all numbers \(x\) that satisfy the given equation. $$ \frac{\log _{9}(13 x)}{\log _{9}(4 x)}=2 $$

Short Answer

Expert verified
The numbers \(x\) that satisfy the given equation are \(x = 0\) and \(x = \frac{13}{16}\).

Step by step solution

01

Given equation

We are given the equation: \[ \frac{\log _{9}(13x)}{\log _{9}(4x)}=2. \]
02

Apply the change of base formula

Applying the change of base formula for logarithms, we can rewrite the equation as: \[ \frac{\log (13x)}{\log (4x)} = 2. \]
03

Eliminate the logarithms

Now, we will cross-multiply and eliminate the logarithms by writing the equation as: \[ \log (13x) = 2 \log (4x). \]
04

Apply the logarithm property

Using the logarithm property, we can rewrite the equation as: \[ \log (13x) = \log( (4x)^2). \]
05

Eliminate the logarithms

Since the logarithms are equal, their arguments must also be equal. We get: \[ 13x = (4x)^2. \]
06

Simplify the equation

We can simplify the equation by expanding the square term: \[ 13x = 16x^2. \]
07

Rearrange the equation

Rearrange the equation to make it a quadratic equation: \[ 16x^2 - 13x = 0. \]
08

Factor the quadratic equation

Factoring the quadratic equation as: \[ x(16x - 13) = 0. \]
09

Find the solutions for x

The solutions for x are: \[ x = 0 \quad \text{or} \quad x = \frac{13}{16}. \] Hence, the numbers \(x\) that satisfy the given equation are \(x = 0\) and \(x = \frac{13}{16}\).

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