Question

Find the value of given logarithmic expression ${\log _{{\pi }}}\tan \left( {0.25{{\pi }}} \right)$.

Answer 1

Hint: To solve this problem we will first find the value of tangent function by converting the radian into degrees, and then solve the logarithmic function. The formula for these are-
$\begin{align}
  &{{\pi }}\;rad = {180^{\text{o}}} \\
  &{{\text{a}}^{\text{b}}} = {\text{c}} \Rightarrow {\text{b}} = {\log _{\text{a}}}{\text{c}} \\
\end{align} $

Complete step-by-step solution -
We will first solve the tangent function by using the given conversion, that is-
$\begin{align}
  & = \tan \left( {0.25 \times 180} \right) \\
  & = \tan {45^o} = 1 \\
\end{align} $
So the expression is-
$\begin{align} \\
  & = {\log _{{\pi }}}\tan \left( {0.25{{\pi }}} \right) \ \\
  & = {\log _{{\pi }}}1\\
 & \text{Let}\;\text{this}\; \text{value}\; \text{be}\; {\text{x}}, \\
 & {\text{x}} = {\log _{{\pi }}}1 \\
 & \text{Using}\; \text{the}\; \text{conversion}, \\
 & {{\pi}^{\text{x}}} = 1 \\
\end{align} $
We know that if the power of any real number is 0, then the result is 1. So, we can write that x = 0
x = 0
${\log _{{\pi }}}\tan \left( {0.25{{\pi }}} \right)=0$
This is the required answer.

Note: It is recommended that we solve the expression taking one function at a time. First solve the innermost function and proceed outwards step by step to get the required answer.