/*! 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 85 Solve $$ \log _{x} 7=\frac{1... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 85

Solve $$ \log _{x} 7=\frac{1}{2} $$

Short Answer

Expert verified
x = 49

Step by step solution

01

Recall the definition of logarithm

To solve the equation \(\text{log}_{x}(7) = \frac{1}{2}\), recall that a logarithm \( \text{log}_{b}(a) \) is the exponent to which the base \( b \) must be raised to produce \( a \).
02

Convert the logarithmic form to exponential form

Using the definition of logarithms, rewrite \( \text{log}_{x}(7) = \frac{1}{2} \) in its exponential form: \ x^{\frac{1}{2}} = 7 \.
03

Solve for the base x

To solve for \( x \), square both sides to eliminate the fraction in the exponent: \ (x^{\frac{1}{2}})^{2} = 7^{2} \. This simplifies to \ x = 49 \.

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Key Concepts

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

Logarithm Definition
Understanding the definition of a logarithm is fundamental to solving logarithmic equations. A logarithm \(\text{log}_{b}(a)\) asks the question: 'To what exponent must the base \({b}\) be raised to produce the number \({a}\)?' For example, \(\text{log}_{2}(8) = 3\) means that 2 raised to the power of 3 gives 8. Thus, the logarithmic equation \(\text{log}_{x}(7) = \frac{1}{2}\) means that \({x}\) raised to the power of \(\frac{1}{2}\) equals 7.
Exponential Form
To solve a logarithmic equation, converting it to an exponential form often simplifies the problem. Using the logarithmic definition, \(\text{log}_{x}(7)= \frac{1}{2}\) can be rewritten in exponential form: \({x}^{\frac{1}{2}} = 7\). This step helps by transforming the logarithm into an equation involving powers, making it easier to solve.
Solving for Base
After converting the equation to its exponential form \({x}^{\frac{1}{2}} = 7\), the next step is to solve for the base \({x}\). We need to eliminate the exponent \(\frac{1}{2}\) to solve for \({x}\). To do this, square both sides: \((x^{\frac{1}{2}})^{2} = 7^{2}\). This simplifies to \({x} = 49\). Now that the exponent has been removed, our solution for the base \({x}\) is 49.

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