# How do you evaluate log_9 0.4 using the change of base formula?

Apr 17, 2018

${\log}_{9} \left(0.4\right) = \ln \frac{0.4}{\ln} \left(9\right) \approx - 0.4170218835$

#### Explanation:

for:

$y = {\log}_{9} \left(0.4\right) \iff {9}^{y} = 0.4$

${9}^{y} = 0.4$

Taking natural logarithms of both sides:

$y \ln \left(9\right) = \ln \left(0.4\right)$

Dividing by $\ln \left(9\right)$

$y = \ln \frac{0.4}{\ln} \left(9\right)$

But:

$y = {\log}_{9} \left(0.4\right)$

$\therefore$

${\log}_{9} \left(0.4\right) = \ln \frac{0.4}{\ln} \left(9\right)$

This is the change of base formula.

${\log}_{9} \left(0.4\right) = \ln \frac{0.4}{\ln} \left(9\right) \approx - 0.4170218835$