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Chapter 4: Problem 54

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{5}(x-7)=2$$

Short Answer

Expert verified
The exact solution for the logarithmic equation is \(x = 32\).

Step by step solution

01

Understand the Basis of Logarithms

A logarithm is an operation that attempts to find out the exponent for a given base to yield the argument. In essence, an equation in the form of \(\log _{b}(a)=c\) is equivalent to \(b^{c}=a\).
02

Convert Logarithmic Equation Into Exponential Form

This equation, \(\log _{5}(x - 7) = 2\), can be written in exponential form as \(5^2 = x - 7\). Hence, we get \(25 = x - 7\).
03

Solve for x

Add 7 to both sides of the equation to solve for x, \(x = 25 + 7\). Thus, \(x = 32\).
04

Check for the Validity of the Solution

Substitute x=32 to the original equation \(\log _{5}(32 - 7)\). It simplifies to \(\log _{5}(25)\), which equals to 2. So, the solution is valid.
05

Provide Exact and Approximate Answers

The exact solution is \(x = 32\). Since it doesn't involve irrational numbers, there's no need for a decimal approximation.

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Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\mathrm{TRACE}]\) and \([\mathrm{ZOOM}]\) features or the intersect command of your graphing utility to verify your answer.
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