# How do you evaluate log_25 5?

We have

${\log}_{25} 5 = \log \frac{5}{\log 25} = \log \frac{5}{\log {5}^{2}} = \log \frac{5}{2 \cdot \log 5} = \cancel{\log} \frac{5}{2 \cdot \cancel{\log} 5} = \frac{1}{2}$

Aug 21, 2016

${\log}_{25} 5 = \frac{1}{2}$

#### Explanation:

You can find the answer by inspection if you understand what question is being asked when you are working in log form.

${\log}_{\textcolor{red}{10}} \textcolor{b l u e}{100}$ asks the question........,

$\text{ What power of "color(red)(10)" will give the number } \textcolor{b l u e}{100}$?"

The answer is obviously 2. Because ${10}^{2} = 100$

$\therefore {\log}_{10} 100 = 2 \mathmr{and} {\log}_{10} 1000 = 3$

Can you see that in the same way....

log_3 27 = 3, and log_5 125 = 3 and log_8 64 = 2 ?

A square root can also be written as an index.

$\sqrt{x} = {x}^{\frac{1}{2}} \text{ " and " } \sqrt{49} = {49}^{\frac{1}{2}} = 7$

${\log}_{25} 5 \text{ asks the question 'what power of 25 will give 5'? }$

As 5 is the square root of 25, the index we need is $\frac{1}{2}$

${\log}_{25} 5 = \frac{1}{2}$

Can you answer these as well?

${\log}_{36} 6 , \text{ " log_81 3, " } {\log}_{32} 2$

$\frac{1}{2} \text{ " 1/4, " } \frac{1}{5}$