/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 105 Solve \(\log _{4}(3 x-2)=2\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 105

Solve \(\log _{4}(3 x-2)=2\)

Short Answer

Expert verified
x = 6.

Step by step solution

01

Understand the Logarithmic Equation

Given the logarithmic equation \(\log_{4}(3x - 2) = 2\), the goal is to solve for x.
02

Rewrite the Equation in Exponential Form

Recall that \(\log_{b}(a) = c\) can be rewritten as \(b^{c} = a\). Apply this to the given equation: \(4^{2} = 3x - 2\).
03

Simplify the Exponent

Calculate \(4^{2}\): 4 squared is 16, so the equation becomes \(16 = 3x - 2\).
04

Isolate the Variable Term

Add 2 to both sides of the equation to isolate the term with the variable: \(16 + 2 = 3x\), which simplifies to \(18 = 3x\).
05

Solve for x

Divide both sides of the equation by 3 to solve for x: \(x = \frac{18}{3}\).
06

Final Answer

Simplify \(\frac{18}{3}\) to get \(x = 6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Logarithms are essential in mathematics, particularly in the context of solving exponential equations. A logarithm answers the question:

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Most popular questions from this chapter

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Solve Is it easier to find \(x\) given \(x=\log _{9} \frac{1}{3}\) or given \(9^{x}=\frac{1}{3} ?\) Explain your reasoning.
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