# How do you use the Change of Base Formula and a calculator to evaluate the logarithm log_pi 14.2?

Dec 30, 2015

${\log}_{\pi} 14.2 \approx 2.32$

#### Explanation:

To calculate ${\log}_{\pi} 14.2$, use any logarithm available on the calculator:

${\log}_{10} 14.2 \approx 1.15$

Then divide the result by the logarithm of the number you wish to use as your base:

${\log}_{10} \pi \approx 0.497$

$\frac{1.15}{0.497} \approx 2.32$

Simplified into a single expression:

${\log}_{10} \frac{14.2}{\log} _ 10 \pi \approx 2.32$

(Note how both logarithms above use the same base. The Change of Base formula will not yield the correct result, otherwise.)

Another common base is base $e$, or the natural logarithm (written $\ln$).

This would be input into a calculator as:

$\frac{\ln \left(14.2\right)}{\ln \left(\pi\right)}$