Question

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

Answer 1

Hint: 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.