# What is the common logarithm of 54.29?

Sep 10, 2015

$\log \left(54.29\right) \approx 1.73472$

#### Explanation:

$x = \log \left(54.29\right)$ is the solution of ${10}^{x} = 54.29$

If you have a natural log ($\ln$) function but not a common $\log$ function on your calculator, you can find $\log \left(54.29\right)$ using the change of base formula:

${\log}_{a} \left(b\right) = {\log}_{c} \frac{b}{\log} _ c \left(a\right)$

So:

$\log \left(54.29\right) = {\log}_{10} \left(54.29\right) = {\log}_{e} \frac{54.29}{\log} _ e \left(10\right) = \ln \frac{54.29}{\ln} \left(10\right)$

Sep 10, 2015

If you are using tables, you need:

#### Explanation:

$\log 54.29 = \log \left(5.429 \times {10}^{1}\right)$

• log(5.429) +1.

From tables

$\log 5.42 = 0.73400$

$\log 5.43 = 0.73480$

$5.429$ is $\frac{9}{10}$ of the way from $5.42 \text{ to } 5.43$, so we get
$\frac{9}{10} = \frac{x}{80}$ so $x = 72$

by linear interpolation,

$\log \left(5.429\right) = 0.74372$

So
$\log \left(54.29\right) = 1.74372$

(I'm using $=$ rather than $\approx$ in each case.)