# How do you calculate log_2 10?

Jun 8, 2016

$x = 3.3219$

#### Explanation:

It helps to write this in another way to understand what is being asked. We can write log form in exponential form
If ${\log}_{2} 10 = x \text{ } \Rightarrow {2}^{x} = 10$

Which power of 2 is equal to 10? It is obviously not a whole number because the powers of 2 are: 2, 4, 8, 16 , 32 .....

So $x$ lies between 3 and 4.

Using index form first, find the log of both sides:

$\log {2}^{x} = \log 10 \text{ } \Rightarrow x \log 2 = \log 10$
$\text{ } x = \log \frac{10}{\log} 2$

We know log 10 = 1, but need to use a calculator to find log2.

Using log form, this can be written using the change of base rule:

${\log}_{2} 10 = x \text{ } x = \log \frac{10}{\log} 2$
This is the same result as was found the first time.

Now use a calculator to find the answer as $x = 3.3219$ which is exactly as we expected, somewhere between 3 and 4.