Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors. $$ \ln \left[\frac{x^{2}-x-2}{(x+4)^{2}}\right]^{1 / 3} \quad x>2 $$

Short answer

The expression simplifies to \(\frac{1}{3} \text{ln}(x^2 - x - 2) - \frac{2}{3} \text{ln}(x+4)\).

Step-by-step solution

Apply the power rule

Use the power rule of logarithms, which states that \(\text{log}_a(b^c) = c \text{log}_a(b)\), to move the exponent \(\frac{1}{3}\) outside the logarithm: \(\text{ln} \left[\frac{x^2 - x - 2}{(x+4)^2}\right]^{1/3} = \frac{1}{3} \text{ln} \left[\frac{x^2 - x - 2}{(x+4)^2}\right]\).

Apply the quotient rule

Use the quotient rule of logarithms, which states that \(\text{log}_a \left(\frac{b}{c}\right) = \text{log}_a(b) - \text{log}_a(c)\), to separate the numerator and the denominator: \(\frac{1}{3} \text{ln} \left[\frac{x^2 - x - 2}{(x+4)^2}\right] = \frac{1}{3} [\text{ln}(x^2 - x - 2) - \text{ln}(x+4)^2]\).

Simplify the expression

Apply the power rule to the \(\text{ln}(x+4)^2\) term: \(\text{ln}(x+4)^2 = 2 \text{ln}(x+4)\). Now the expression becomes \(\frac{1}{3} [\text{ln}(x^2 - x - 2) - 2 \text{ln}(x+4)]\).

Distribute the fraction

Distribute \(\frac{1}{3}\) to each term inside the brackets: \(\frac{1}{3} \text{ln}(x^2 - x - 2) - \frac{2}{3} \text{ln}(x+4)\).

Key concepts

Logarithm Rules

This rule is crucial when dealing with exponents. It allows for moving the exponent in front of the log, simplifying calculations. By understanding and using these fundamental rules, you can make solving logarithmic expressions a breeze.

Power Rule

This step makes subsequent simplification easier. Remember, the power rule is handy whenever you see an exponent inside the logarithm.

Logarithmic Simplification

By now, we've simplified the complicated initial expression into a more handable form. Understanding how to apply these rules systematically allows you to simplify various logarithmic expressions.