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Chapter 3: Problem 70

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
The solution to the equation is approximately \(x = 0.404\).

Step by step solution

01

Rewrite the Equation

First, make use of the property of logs to rewrite the equation. Combine the two logarithms on the left and right side of the equation into one: \(\ln\frac{x+1}{x} = 2\)
02

Solve for x

The equation can now be rewritten in exponential form as \(\frac{x+1}{x} = e^2\). Solving it for x would give us the value of x as \(x = \frac{1}{e^2 - 1}\)
03

Decimal Approximation

Approximate the value of x up to three decimal points. This would give \(x = 0.404\)
04

Verify Result

The final step is to substitute the found solution into the original equation to verify if it is indeed the solution. Doing this, we find that the left side and right side of the equation do equate, verifying our solution.

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