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Chapter 3: Problem 68

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$10-4 \ln (x-2)=0$$

Short Answer

Expert verified
The result derived from both algebraic calculation and graphing utility is that \(x\) approximately equals to \(e^{5/2} + 2\), which is about 6.500.

Step by step solution

01

Isolate the logarithm

First, isolate the logarithm by adding \(4\ln(x-2)\) to both sides of the equation. This will yield: \(4\ln(x-2)=10\)
02

Convert to exponential form

Next, convert the equation from logarithmic form to exponential form. The base of the natural logarithm is e, so you can rewrite the equation as: \(e^{10/4}=x-2\)
03

Solve for x

Next, solve for x by adding 2 to both sides of the equation. This will yield: \(x=e^{10/4}+2\)
04

Use the graphing utility

Plug the equation \(10-4\ln(x-2)\) into the graphing utility and find the x-intercept. This should yield a value of x that is close to the value found algebraically.
05

Verify the result

Finally, verify the result by substituting the found x value back into the original equation. The left hand side should equal to the right hand side after the substitution.

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Most popular questions from this chapter

Writing a Natural Logarithmic Equation In Exercises \(53-56,\) write the exponential equation in logarithmic form. $$e^{1 / 2}=1.6487 \ldots$$

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