The discussion focuses on the simplification of the logarithmic expression $\displaystyle \frac{1}{5}\ln|\sin5x|+\frac{1}{5}\ln|\csc5x-\cot5x|$. Participants explore various methods and steps involved in simplifying this expression, including factoring and combining logarithms.
Participants generally agree on the steps to simplify the expression, but there are variations in the final forms presented and whether the modulo sign can be removed. No consensus is reached on a single final expression.
Some steps in the simplification process may depend on specific assumptions about the values of $x$ and the behavior of the trigonometric functions involved.
Ackbach said:What have you tried?
\begin{array}{ccc}\text{Factor:} & \tfrac{1}{5}\big(\ln|\sin5x| + \ln|\csc5x - \cot5x|\big) \\ \\ \text{Combine logs:} & \tfrac{1}{5}\ln|\sin5x(\csc5x - \cot5x)| \\ \\ \text{Distribute:} & \tfrac{1}{5}\ln|\sin5x\csc5x - \sin5x\cot5x| \\ \\ \text{Simplify:} & \tfrac{1}{5}\ln|1-\cos5x| \\ \\ \text{Further?} & \tfrac{1}{5}\ln\left|2\sin^2\! \tfrac{5x}{2}\right| \end{array}\text{Simplify: }\:\tfrac{1}{5}\ln|\sin5x|+\tfrac{1}{5}\ln|\csc5x-\cot5x|
soroban said:Hello, paulmdrdo!
\begin{array}{ccc}\text{Factor:} & \tfrac{1}{5}\big(\ln|\sin5x| + \ln|\csc5x - \cot5x|\big) \\ \\ \text{Combine logs:} & \tfrac{1}{5}\ln|\sin5x(\csc5x - \cot5x)| \\ \\ \text{Distribute:} & \tfrac{1}{5}\ln|\sin5x\csc5x - \sin5x\cot5x| \\ \\ \text{Simplify:} & \tfrac{1}{5}\ln|1-\cos5x| \\ \\ \text{Further?} & \tfrac{1}{5}\ln\left|2\sin^2\! \tfrac{5x}{2}\right| \end{array}