# Based on the estimates log(2) = .03 and log(5) = .7, how do you use properties of logarithms to find approximate values for log_5(2)?

##### 1 Answer
May 28, 2015

I woul change base of the logarithm using the property:
${\log}_{b} x = \frac{{\log}_{a} x}{{\log}_{a} \left(b\right)}$
in your case:
${\log}_{5} \left(2\right) = \frac{\log 2}{\log \left(5\right)} = \frac{0.03}{0.7} = \frac{3}{100} \cdot \frac{10}{7} = \frac{3}{70} = 0.043$

Although I am not completely sure about your estimates (in particular 0.03)....the result should give you $0.43$ so that ${5}^{0.43} = 2$!