/*! 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 80 Solve $$ \log _{4} x=3 $$... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 80

Solve $$ \log _{4} x=3 $$

Short Answer

Expert verified
The solution is \(x = 64\).

Step by step solution

01

Understand the Given Logarithm

The given logarithmic equation is \(\log_{4} x = 3\). This means that the logarithm of \(x\) to the base 4 equals 3.
02

Convert the Logarithmic Equation to Exponential Form

Recall that \(\log_{b} a = c\) is equivalent to \(b^c = a\). Therefore, \(\log_{4} x = 3\) can be written as \(4^3 = x\).
03

Simplify the Exponential Expression

Calculate \(4^3\): \(4^3 = 4 \times 4 \times 4 = 64\). Thus, \(x = 64\).
04

Verify the Solution

Double-check the solution by substituting \(x = 64\) back into the original equation: \(\log_{4} 64 = 3\). Since \(4^3 = 64\), the solution is correct.

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. They help us understand how numbers are related in terms of exponentiation. For a logarithm with base 4, like \(\log_{4} x\), we are looking for the power to which 4 must be raised to give us the number inside the log, which in this case is x. Hence, \(\log_{4} x = 3\) means 4 raised to the power of 3 equals x. Understanding this inverse relationship is crucial for solving logarithmic equations.
Exponential Form
Converting logarithms to their exponential form can simplify the solving process. For a logarithmic equation \(\log_{b} a = c\), it can be rewritten in exponential form as \(\ b^c = a\). In our exercise, \(\log_{4} x = 3\) translates to \(\4^3 = x\). This translation is key because it often makes it easier to solve for x. Calculating \(\4^3\), we get 64, so x equals 64.
Verification of Solutions
Verification ensures our solution is correct. By substituting x back into the original equation, we double-check our work. For \(\x = 64\), we substitute back into \(\log_{4} x = 3\). This becomes \(\log_{4} 64\). Since we know \(\4^3 = 64\), this confirms that \(\log_{4} 64 = 3\), proving our solution is accurate. Always verify to avoid errors.

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