# How do you evaluate log_625 25?

Sep 1, 2016

${\log}_{625} 25 = \textcolor{g r e e n}{\frac{1}{2}}$

#### Explanation:

Note that $625 = {25}^{2} \textcolor{w h i t e}{\text{XX}} \rightarrow 25 = {625}^{\frac{1}{2}}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} {\log}_{625} 25 = {\log}_{625} {626}^{\frac{1}{2}}$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow {\log}_{625} 25 = \frac{1}{2}$ (by definition of log)

Sep 1, 2016

${\log}_{625} 25 = \frac{1}{2}$

#### Explanation:

Logs are easier to understand if you think about an expression given in log form as asking a question.

In ${\log}_{625} 25$, the question being asked is;

"What power/index of 625 will give 25?"
OR " How do I make 625 into 25?"

You should recognise $25$ as being the square root of 625.

A square root can be written as an index.

$\sqrt{x} = {x}^{\frac{1}{2}}$

$\sqrt{625} = {625}^{\frac{1}{2}} \leftarrow$ this answers our question!

${\log}_{625} 25 = \frac{1}{2}$

Log form and index form are interchangeable.

${\log}_{a} b = c \Leftrightarrow {a}^{c} = b$

${\log}_{625} 25 = x \Leftrightarrow {625}^{x} = 25$

${625}^{x} = {\left({25}^{2}\right)}^{x} = {25}^{1} \leftarrow$ make the bases the same

${25}^{2 x} = {25}^{1} \leftarrow$ equate the indices and solve for x

$2 x = 1 \rightarrow x = \frac{1}{2}$