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Chapter 3: Problem 67

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right]$$

Short Answer

Expert verified
The condensed logarithmic expression of the given expression is \(\ln \left[\left(\frac{x^{2} + 10x + 25}{x^{3} - 4x}\right)^{1/3}\right].\)

Step by step solution

01

Apply the first Logarithmic Property

Rewrite the expression applying \(a \ln m = \ln m^{a}\): \[\frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right] = \frac{1}{3}\left[\ln (x+5)^{2}-\ln x-\ln \left(x^{2}-4\right)\right].\]
02

Apply the second Logarithmic Property

Further rewrite the expression applying \(\ln a - \ln b = \ln(a/b)\): \[\frac{1}{3}\left[\ln (x+5)^{2}-\ln x-\ln \left(x^{2}-4\right)\right] = \frac{1}{3}\left[\ln \frac{(x+5)^{2}}{x \left(x^{2}-4\right)}\right].\]
03

Simplify the expression inside the logarithm

Simplify the fraction: \[\frac{(x+5)^{2}}{x \left(x^{2}-4\right)} = \frac{(x+5)^{2}}{x^{3}-4x} = \frac{x^{2} + 10x + 25}{x^{3} - 4x}\], so our expression becomes: \[\frac{1}{3}\left[\ln \frac{x^{2} + 10x + 25}{x^{3} - 4x}\right].\]
04

Apply the last Logarithmic Property

Last step is to apply \(a \ln m = \ln m^{a}\) again: \[\frac{1}{3}\left[\ln \frac{x^{2} + 10x + 25}{x^{3} - 4x}\right] = \ln \left[\left(\frac{x^{2} + 10x + 25}{x^{3} - 4x}\right)^{1/3}\right].\]

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