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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 for the equation is \(x = 1\).

Step by step solution

01

Combine logarithms

Utilize the subtraction property of logarithms to combine them: \( \ln 3 - \ln(x+5) - \ln x = \ln \left( \frac{3}{(x+5)x} \right) = 0 \)
02

Use definition of logarithm

Recall that \(\ln a = b\) is equivalent to \(e^b = a\). Thus, the equation becomes: \( e^0 = \frac{3}{(x+5)x} \)
03

Simplify and solve for x

Solving \( e^0 = \frac{3}{(x+5)x} \), we get \(1 = \frac{3}{(x+5)x}\). Clearing the fraction gives \(x^2 + 5x - 3 = 0\). Factoring this quadratic equation gives \((x+3)(x-1) = 0\). Therefore, the possible solutions are \(x = -3\) and \(x = 1\)
04

Check validity of the solutions

We can't use \(x = -3\) in the original equation because it would result in a negative argument to the natural logarithm, which is undefined. Therefore, the only valid solution is \(x = 1\) .

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I cannot simplify the expression \(b^{m}+b^{n}\) by adding exponents, there is no property for the logarithm of a sum.

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.

Explaining the Concepts Suppose that a population that is growing exponentially increases from \(800,000\) people in 2010 to \(1,000,000\) people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

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You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.
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