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Chapter 4: Problem 100

Solve each equation. $$\ln 3-\ln (x+5)-\ln x=0$$

Short Answer

Expert verified
The solution to the equation is \( x = \frac{-5 + \sqrt{37}}{2} \).

Step by step solution

01

Combine Logarithms

Utilize the properties of logarithms to combine the logarithm terms. According to the properties, \( \ln a - \ln b = \ln \frac{a}{b}\). Therefore, the given equation \( \ln 3 - \ln (x+5) - \ln x = 0 \) becomes \( \ln 3 - \ln ((x+5)x) = 0 \) or \( \ln \frac{3}{(x+5)x} = 0\).
02

Convert Logarithmic Equation to Exponential Form

The property of logarithms states that \( \ln a = b \) is equivalent to \( e^b = a \). Therefore, convert the logarithmic equation \( \ln \frac{3}{(x+5)x} = 0 \) to the exponential form \( e^0 = \frac{3}{(x+5)x} \). We know that \( e^0 = 1 \), thus the equation simplifies to \( \frac{3}{(x+5)x} = 1 \).
03

Solve for x

Cross-multiply and set the equation equal to zero to solve for x. This gives us \( 3 = (x+5)x \), which simplifies to \( x^2 + 5x - 3 = 0 \). This is a quadratic equation, and we can solve for x using the quadratic formula \( x = {-b 卤 \sqrt{b^2 - 4ac}}/{2a} \). Substituting \( a = 1, b = 5, c = -3 \) gives us \( x = \frac{-5 卤 \sqrt{5^2 - 4*1*-3}}{2*1} \), which simplifies to \( x = \frac{-5 卤 \sqrt{25 + 12}}{2} \) or \( x = \frac{-5 卤 \sqrt{37}}{2} \).
04

Check for extraneous solutions

Logarithms are only defined for positive numbers, so we need to check if the solutions \( x = \frac{-5 + \sqrt{37}}{2} \) and \( x = \frac{-5 - \sqrt{37}}{2} \) are valid. Since \( \frac{-5 - \sqrt{37}}{2} \) is negative, which makes \( \ln (x+5) \) undefined, it is not valid. The others solution is \( x = \frac{-5 + \sqrt{37}}{2} \), which is positive and within the range for which logarithms are defined.

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Most popular questions from this chapter

This group exercise involves exploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) \(7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.

Describe the following property using words: \(\log _{b} b^{x}=x\).

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1), \quad\) where \(\quad 0 \leq t \leq 12 . \quad\) Use \(\quad\) a graphing utility to graph the function. Then determine how many months elapsed before the average score fell below 65.

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