# How do you calculate log_9 14 with a calculator?

Sep 27, 2016

$1.201$

#### Explanation:

Log form and index form are interchangeable.

${\log}_{a} b = c \text{ " hArr " } {a}^{c} = b$

Let ${\log}_{9} 14 = x \text{ } \rightarrow {9}^{x} = 14$

As $x$ is in the index, use logs to solve.

$\log {9}^{x} = \log 14 \text{ } \leftarrow$ log power law

$x \log 9 = \log 14 \text{ "larr" isolate } x$

$x = \log \frac{14}{\log} 9 \text{ } \leftarrow$ use a calculator.

(if no base is given, it is implied it is base 10)

$x = 1.201$

The same result would have been obtained from using the "Change of base law"

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a} \text{ } \leftarrow$ (c can be any base)

${\log}_{9} 14 = \frac{{\log}_{10} 14}{{\log}_{10} 9}$

$= 1.201$