# Logarithm concepts -- introduction

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This module introduces the concept of logarithms.

Suppose you are a biologist investigating a population that doubles every year. So if you start with 1 specimen, the population can be expressed as an exponential function: $p\left(t\right)={2}^{t}$ where $t$ is the number of years you have been watching, and $p$ is the population.

Question: How long will it take for the population to exceed 1,000 specimens?

We can rephrase this question as: “2 to what power is 1,000?” This kind of question, where you know the base and are looking for the exponent, is called a logarithm .

${\text{log}}_{2}\text{1000}$ (read, “the logarithm, base two, of a thousand”) means “2, raised to what power, is 1000?”

In other words, the logarithm always asks “ What exponent should we use ?” This unit will be an exploration of logarithms.

## A few quick examples to start things off

 Problem Means The answer is because ${\text{log}}_{2}8$ 2 to what power is 8? 3 ${2}^{3}$ is 8 ${\text{log}}_{2}16$ 2 to what power is 16? 4 ${2}^{4}$ is 16 ${\text{log}}_{2}10$ 2 to what power is 10? somewhere between 3 and 4 ${2}^{3}=8$ and ${2}^{4}=16$ ${\text{log}}_{8}2$ 8 to what power is 2? $\frac{1}{3}$ ${8}^{\frac{1}{3}}=\sqrt[3]{8}=2$ ${\text{log}}_{10}10,000$ 10 to what power is 10,000? 4 ${10}^{4}=10,000$ ${\text{log}}_{10}\left(\frac{1}{100}\right)$ 10 to what power is $\frac{1}{100}$ ? –2 ${10}^{–2}=\frac{1}{{10}^{2}}=\frac{1}{100}$ ${\text{log}}_{5}0$ 5 to what power is 0? There is no answer ${5}^{\text{something}}$ will never be 0

As you can see, one of the most important parts of finding logarithms is being very familiar with how exponents work!

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