Problem

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Evaluate logarithmic expressions if possible. \[ 5 \ln (x+8)-5 \ln x \] \[ 5 \ln (x+8)-5 \ln x=\square \]

Solution

Step 1 :Given the expression \(5 \ln (x+8)-5 \ln x\)

Step 2 :We can use the properties of logarithms to condense the expression into a single logarithm. The property we will use is \(\ln a - \ln b = \ln \left(\frac{a}{b}\right)\)

Step 3 :Applying this property to the given expression, we get \(\ln \left(\frac{(x+8)^5}{x^5}\right)\)

Step 4 :This is the simplified form of the given expression. The coefficient of the logarithm is 1, as required. The expression cannot be evaluated further without a specific value for x.

Step 5 :\(\boxed{\ln \left(\frac{(x+8)^5}{x^5}\right)}\) is the final answer.

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Source: https://solvelyapp.com/problems/flSri1JbBr/

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