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Chapter 3: Problem 56

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

Short Answer

Expert verified
The solution to the logarithmic equation approximately to three decimal places is \( x \approx 0.352 \).

Step by step solution

01

Combine the Logarithms

Use the properties of logarithms to combine the two natural logs into one. According to the properties of logs, \( \ln a + \ln b = \ln (a \cdot b) \). So, the equation becomes \( \ln (x \cdot (x + 1)) = 1 \).
02

Exponentiate Both Sides

Remove the logarithm by exponentiating both sides of the equation. We get \( e^{\ln (x \cdot (x + 1))} = e^1 \), which simplifies to \( x \cdot (x + 1) = e \).
03

Solve the Quadratic Equation

Rewrite the equation in general form to solve. We have \( x^2 + x - e = 0 \). This is a quadratic equation, and we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find x.
04

Apply the Quadratic Formula

By applying the quadratic formula, we find two solutions. The solutions are \( x = \frac{-1 + \sqrt{1 + 4e}}{2} \) and \( x = \frac{-1 - \sqrt{1 + 4e}}{2} \).
05

Select the Valid Solution

Since \( x \) must be greater than 0 for the logarithm to exist, we reject the negative solution. Thus the solution is \( x = \frac{-1 + \sqrt{1 + 4e}}{2} \).
06

Approximate the Result

Approximate this value to three decimal places using a calculator, they get \( x \approx 0.352 \).

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Most popular questions from this chapter

In Exercises \(97-102,\) determine whether the statement is true or false given that \(f(x)=\ln x .\) Justify your answer. $$f(x-2)=f(x)-f(2), \quad x > 2$$

Use a graphing utility to graph and solve the equation. Approximate the result to three decimal places. Verify your result algebraically. $$8 e^{-2 x / 3}=11$$

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-4}$$

Solve the equation algebraically. Round your result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$
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