Problem 131 - College Algebra Essentials

Question

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$

Step-by-Step Solution

Answer

The simplified single term form of the given expression is \(4x^{3}\).

Step 1: Identify the properties of logarithms

From logarithmic properties, it is known that \(\ln a - \ln b = \ln(a/b)\). This can be used to simplify the expression in the exponent.

Step 2: Simplify the exponent

Applying the logarithmic property to the exponent gives \(e^{\ln(8x^{5}/2x^{2})} = e^{\ln(4x^{3})}\).

Step 3: Further Simplify the expression

From the property of exponential and logarithm, it is known that \(e^{\ln a} = a\). Apply that to our simplified expression from step 2 : \(e^{\ln(4x^{3})} = 4x^{3}\).