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Chapter 3: Problem 37

Find all numbers \(x\) that satisfy the given equation. $$ \frac{\log _{6}(15 x)}{\log _{6}(5 x)}=2 $$

Short Answer

Expert verified
The solution to the given equation is \(x = 0\) or \(x = \frac{3}{5}\).

Step by step solution

01

Understanding the equation

We have the equation: \(\frac{\log_{6}(15x)}{\log_{6}(5x)} = 2\) We know that the base of both logarithms is 6, and we also know that the change of base formula for logarithms states: \( \frac{\log_{a}{b}}{\log_{a}{c}} = \log_{c}{b} \)
02

Applying the change of base formula

Applying the change of base formula to our equation, we get: \(\log_{5x}(15x) = 2\) Which means: \((5x)^2 = 15x\)
03

Expanding and simplifying the equation

We expand the equation and simplify: \(25x^2 - 15x = 0\) Now, factor out \(5x\): \(5x(5x - 3) = 0\)
04

Solving for x

We have two factors, so we set each one to 0 and solve for \(x\): Case 1: \(5x = 0\) \(x = 0\) Case 2: \(5x - 3 = 0\) \(5x = 3\) \(x = \frac{3}{5}\) So the solution to the given equation is: \(x = 0\) or \(x = \frac{3}{5}\)

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