# How do you evaluate log_(4) 66 ?

Jun 12, 2016

$x = \log \frac{66}{\log} 4$

Using a calculator gives $x = 3.022$

#### Explanation:

The answer is not going to be an integer. The question is asking:
"To what power must 4 be raised, to equal 66?"

The powers of 4 are : 4, 16, 64, 256 ....
66 is not one of them, but we can see that the required index will be 3. .....
log from is interchangeable with index form:

If ${\log}_{4} 66 = x \text{ then } {4}^{x} = 66$

Let's use the index form and find the log of both sides, because the variable is in the index.
$\log {4}^{x} = \log 66$

$x \log 4 = \log 66$

$x = \log \frac{66}{\log} 4$

Using a calculator gives $x = 3.022$

This same result could be found from the change of base law:

${\log}_{a} b = {\log}_{10} \frac{b}{\log} _ 10 a$

Some calculators can work with any base, but for those that only use base 10, this works fine.1