Question

How do you evaluate ${\log _\pi }e?$

Answer 1

In this type of question we will use logarithmic and exponential base change properties to solve the question. According to base change formula the base of a logarithmic function can be changed as ${\log _a}x = \dfrac{{{{\log }_c}x}}{{{{\log }_c}a}}$ , where $a$ and $x$ are all positive real numbers. Use the above base change formula and we get the required answer.

Complete step by step answer:
Given ${\log _\pi }e$
We use the formula of logarithmic function we get
${\log _\pi }e = \dfrac{{{{\log }}e}}{{{{\log }}\pi }}$
Putting the values of numerator and denominator , we get
$\Rightarrow {\log _\pi }e = \dfrac{{0.4342944819}}{{0.49714987269}}$
Calculating and simplifying in the above equation and we get
$\Rightarrow {\log _\pi }e = 0.873568526$
We take approximation of the above value and we get
$\Rightarrow {\log _\pi }e = 0.87357$ (approximately)

Note: We should use the value of ${\log _e}\pi = 0.49714987269$ and ${\log _e}e = 0.4342944819$ in the function to find the value of $y$. We know the power rule of logarithm $\log ({a^b}) = b.\log a$, the quotient rule of logarithm $\log \left( {\dfrac{a}{b}} \right) = \log a - \log b$ and the product rule of logarithm $\log \left( {a.b} \right) = \log a + \log b$. We use these depending on the problem and we should know that while applying these laws the base of the logarithm should be the same.