Problem 121 - Introductory and Intermediate Algebra for College Students 4th

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 form of the expression \(e^{\ln 8 x^{5}-\ln 2 x^{2}}\) is \(4x^{3}\).

Step 1: Application of Difference of Logarithms Rule

Use the rule for difference of logarithms to combine \(\ln 8 x^{5} - \ln 2 x^{2}\) into \(\ln ((8 x^{5})/(2 x^{2}))\). This simplifies to \(\ln (4x^{3})\). So the expression becomes \(e^{\ln 4x^{3}}\).

Step 2: Application of Exponential Law

Then, apply the exponential law to the expression \(e^{\ln 4x^{3}}\). Per the law \(e^{\ln a} = a\), the expression simplifies to \(4x^{3}\).