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Chapter 3: Problem 109

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{4} x-\log _{4}(x-1)=\frac{1}{2}$$

Short Answer

Expert verified
The algebraic solution to the logarithmic equation, approximated to three decimal places, is \( x = 2.000 \).

Step by step solution

01

Use the property of logarithms

Combine the two logarithms on the left-hand side of the equation into a single logarithm using the property \(\log _{b} a-\log _{b} c=\log _{b} \frac{a}{c}\). The equation becomes: \( \log _{4} \frac{x}{x-1} = \frac{1}{2} \)
02

Eliminate the logarithm

Eliminate the logarithm by raising both sides of the equation to the base of the log. This gives: \( 4^{\left( \log _{4}\frac{x}{x-1}\right)} = 4^{\frac{1}{2}} \). This simplifies to: \( \frac{x}{x-1}=\sqrt{4} \) to give a standard equation without any logs.
03

Solve for x

Simplify the equation further by multiplying both sides by \( x - 1 \). This gives: \( x = 2x - 2 \). Solving for \( x \) results in: \( x = 2 \).
04

Validation

Insert the result into the original equation to check for the validity: \( \log _{4} (2) - \log _{4} (2-1) = \frac{1}{2} \). This simplifies to \( \frac{1}{2} = \frac{1}{2} \), thus the solution is valid.

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Most popular questions from this chapter

Use the following information for determining sound intensity. The level of sound \(\boldsymbol{\beta}\), in decibels, with an intensity of \(I\), is given by \(\boldsymbol{\beta}=10 \log \left(I / I_{0}\right),\) where \(I_{0}\) is an intensity of \(10^{-12}\) watt per square meter, corresponding roughly to the faintest sound that can be heard by the human ear. In Exercises 65 and 66 , find the level of sound \(\boldsymbol{\beta}\). Due to the installation of noise suppression materials, the noise level in an auditorium was reduced from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials.

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Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$3-\ln x=0$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\log _{3} x+\log _{3}(x-8)=2$$
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