# Question #9b04c

Aug 5, 2017

${\log}_{10} \left(1000\right) = 3$

#### Explanation:

I'm assuming that you mean log with base 10. The answer will be different if the base is not 10.

Here are two ways of writing the same expression:

$y = {\log}_{10} \left(x\right)$

and

${10}^{y} = x$

$y = \text{the power}$

$10 = \text{the base}$

$x = \text{the answer}$

We need to solve for the power, $y$. Start with the log form:

$y = {\log}_{10} \left(1000\right)$

Rewrite this expression in index form:

${10}^{y} = 1000$

Rewrite the expression with the same base on both sides:

${10}^{y} = {10}^{3}$

Because we have the same base, the powers must be equal:

$y = 3$

So ${\log}_{10} \left(1000\right) = 3$

See if you can find another way of solving this problem using the following log laws:

${\log}_{a} \left({b}^{n}\right) = n {\log}_{a} \left(b\right)$

${\log}_{a} \left(a\right) = 1$

Aug 5, 2017

$\log \left(1000\right) = 3$

#### Explanation:

Usually $\log$ stands for "logarithm base 10" or ${\log}_{10}$

A logarithm is a function that returns the power to which you must take the base to get the input.

Or in the case of base 10

$\log \left({10}^{x}\right) = x$

and for any base $b$

${\log}_{b} \left({b}^{x}\right) = x$

Since $1000 = 100 \times 10$ and $100 = 10 \times 10$,

then $1000 = 10 \times 10 \times 10$

And by the definition of an exponent

$1000 = 10 \times 10 \times 10 = {10}^{3}$

Then

$\log \left(1000\right) = \log \left({10}^{3}\right) = 3$