# How do you calculate log_2 512?

May 1, 2018

${\log}_{2} \left(512\right) = 9$

#### Explanation:

Notice that 512 is ${2}^{9}$.
$\implies {\log}_{2} \left(512\right) = {\log}_{2} \left({2}^{9}\right)$
By the Power Rule, we may bring the 9 to the front of the log.
$= 9 {\log}_{2} \left(2\right)$

The logarithm of a to the base a is always 1. So ${\log}_{2} \left(2\right) = 1$
$= 9$

the value of ${\log}_{2} 512 = 9$

#### Explanation:

we need to calculate ${\log}_{2} \left(512\right)$
$512 = {2}^{9} \Rightarrow {\log}_{2} \left(512\right) = {\log}_{2} \left({2}^{9}\right)$
${\log}_{a} {b}^{n} = n {\log}_{a} b$ $\Rightarrow {\log}_{2} {2}^{9} = 9 {\log}_{2} 2$
since ${\log}_{a} a = 1 \Rightarrow {\log}_{2} 512 = 9$

May 1, 2018

${\log}_{2} 512 = 9 \text{ }$ because ${2}^{9} = 512$

#### Explanation:

Powers of numbers can be written in index form or log form.
They are interchangeable.

${5}^{3} = 125$ is index form: It states that $5 \times 5 \times 5 = 125$

I think of log form as asking a question. In this case we could ask:

"Which power of $5$ is equal to 125?"
or
"How can I make $5$ into $125$ using an index?"

log_5 125 =?

We find that ${\log}_{5} 125 = 3$

Similarly:
${\log}_{3} 81 = 4 \text{ }$ because ${3}^{4} = 81$
${\log}_{7} 343 = 3 \text{ }$ because ${7}^{3} = 343$

In this case we have:

${\log}_{2} 512 = 9 \text{ }$ because ${2}^{9} = 512$

The powers of $2$ are:

$1 , 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , 512 , 1024$

(From ${2}^{0} = 1$ up to ${2}^{10} = 1024$)

There is a real advantage in learning all the powers up to $1000$, there are not that many and knowing them will make your work on logs and exponential equations SO much easier.