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Chapter 3: Problem 99

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$\ln x-\ln (x+1)=2$$

Short Answer

Expert verified
The solution to the equation \(\ln x- \ln(x+1) = 2\) is approximately \(x 鈮� 7.390\).

Step by step solution

01

Apply the Quotient Rule for Logarithms

The quotient rule states that the difference of two logarithms is equal to the logarithm of the quotient of the two numbers. Applying this to our equation would give us \(\ln (x / (x+1))=2\).
02

Convert the Logarithmic Equation to an Exponential Equation

Next, we turn the logarithmic equation into an equivalent exponential equation using the principle \(b = \ln a\) is equivalent to \(e^b = a\). This changes our equation to \(e^2 = x / (x+1)\).
03

Solve for x

First, we'll clear the fraction. Cross multiplying gives us \(e^2 (x+1) = x\). This simplifies to \(e^2 x + e^2 = x\). Combine like terms and move everything to one side of the equation, giving \(e^2 x -x = -e^2\). Factoring out x, we get \(x(e^2 -1) = -e^2\). Finally, we'll solve for x by dividing both sides by \(e^2-1\), which gives \(x = {-e^2}/ {(e^2-1)}\).
04

Check for Extraneous Solutions

Since we cannot take a logarithm of a non-positive number (0 or negative numbers), we must be sure our solution x is greater than -1 which in our case, it is. So the solution is valid.
05

Round the Answer

Substituting \(e\) as approximately 2.718, and calculating, we have \(x 鈮� 7.390\). As a final answer, round the decimal to three places to get \(x 鈮� 7.390\).

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