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Chapter 3: Problem 116

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$\ln (x+1)=2-\ln x$$

Short Answer

Expert verified
After working through the algebra, you'll find that \(x \approx 0.197\). Graphing the function \(f(x) = \ln (x+1) - 2 + \ln x\) will confirm this result.

Step by step solution

01

Rewrite the equation

Combine the two logarithmic terms into one by subtracting \(\ln x\) from both sides, yielding \(\ln(x+1) - \ln x = 2\).
02

Applying Logarithmic Property

For any pair of positive numbers a and b, we have \(\ln a - \ln b = \ln(a / b)\). Hence \(\ln(x+1) - \ln x\) becomes \(\ln ((x + 1) / x) = 2\).
03

Convert to exponential form

Convert the logarithmic equation to its equivalent exponential form. We know \(a = \ln b\) is equivalent to \(e^a = b\). Hence the equation \(\ln ((x + 1) / x) = 2\) becomes \(e^2 = ((x + 1) / x)\). Simplifying this gives \(x = (x + 1) / e^2\).
04

Solve for x

To solve the equation \(x = (x + 1) / e^2\), multiply both sides of the equation by \(e^2\) and then rearrange terms, which gives \(e^2x - x = 1\). Solving for \(x\) yields \(x = 1 / (e^2 - 1)\).
05

Approximate the result

Substitute the numerical value of \(e\) into \(x = 1 / (e^2 - 1)\) and approximate the result to three decimal places. \(x \approx 0.197\).
06

Verify the result graphically

Use a graphing utility to graph the function \(f(x) = \ln (x+1) - 2 + \ln x\). You should observe that the function intersects the x-axis around \(x = 0.197\), which confirms the result obtained algebraically.

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