# How do you simplify log_10 5?

Sep 14, 2015

It's not really possible to 'simplify' it, but it can be helpful to note:

${\log}_{10} 5 = 1 - {\log}_{10} 2 \approx 1 - 0.30103 = 0.69897$

#### Explanation:

${\log}_{10} \left(5\right) = {\log}_{10} \left(\frac{10}{2}\right) = {\log}_{10} \left(10\right) - {\log}_{10} \left(2\right)$

$= 1 - {\log}_{10} \left(2\right)$

So if you know ${\log}_{10} \left(2\right) \approx 0.30103$, then you don't need to memorise ${\log}_{10} \left(5\right) \approx 1 - 0.30103 = 0.69897$

You can also use the change of base formula:

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a}$

to express ${\log}_{10} \left(2\right)$ in terms of natural logarithms:

${\log}_{10} \left(2\right) = {\log}_{e} \frac{2}{\log} _ e \left(10\right) = \ln \frac{2}{\ln} \left(10\right)$