Problem 131 - Algebra and Trigonometry

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

Step 1: Apply Logarithmic Properties

First, apply the logarithmic property of difference of logs to simplify \( e^{\ln 8 x^{5}-\ln 2x^{2}} \) into \( e^{\ln \frac{8x^5}{2x^2}} \).

Step 2: Simplify the Fraction

Simplify the fraction to obtain \( e^{\ln \frac{4x^3}{1}} \) or \( e^{\ln 4x^3} \).

Step 3: Cancel Out Exponential and Natural Logarithm

Since \( e^x \) and \( \ln x \) are inverse operations, \( e \) and \( \ln \) will cancel out each other. Hence, \( e^{\ln 4x^3} \) simplifies to \( 4x^3 \).