# How do you evaluate log 100?

May 19, 2015

$\log 100 = 2$

To evaluate this you use the definition of a logarithm.
${\log}_{a} b = c \iff {a}^{c} = b$

You also have to assume that if no base $b$ is written then the base is $10$

So in the example you have ${\log}_{10} 100 = c \iff {10}^{c} = 100$.

Now you can easily find that $c = 2$, because ${10}^{2} = 10 \cdot 10 = 100$.

$\log 100 = {\log}_{10} 100 = 2$

Jun 11, 2017

$\log 100 = 2$

#### Explanation:

$\log 100 = 2$ because it can be denoted as

$\log 100 = \log \left(10 \cdot 10\right) = \log 10 + \log 10$

Keep in mind that when there is no base written, it is assumed to be a base of $10$. Applying the rule of $\log x = 1$, you have $\log 10 = 1$.

Then that means

$\log 10 + \log 10 = 1 + 1 = 2$

Also, you can use the power rule.

$\log 100 = \log \left({10}^{2}\right) = 2 \cdot \log 10$

which is equal to

$2 \cdot 1 = 2$

Jun 12, 2017

${10}^{2} = 100 , \text{ } \therefore {\log}_{10} 100 = 2$

#### Explanation:

It might help to compare the two notations:

Index form and log form.$\text{ }$ They are interchangeable.

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

Index form makes a statement:

${10}^{1} = 10 , \text{ " 10^2 = 100," } {10}^{3} = 1000$

log_10 100 =???

"What power of $10$ will give $100$?

${\log}_{10} 100 = 2$

In the same way..

${\log}_{4} 16 = 2 \text{ because } {4}^{2} = 16$

${\log}_{3} 27 = 3 \text{ because } {3}^{3} = 27$

${\log}_{2} 32 = 5 \text{ because } {2}^{5} = 32$