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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: $p(t)={2}^{t}$ 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.
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? | $\frac{1}{3}$ | ${8}^{\frac{1}{3}}=\sqrt[3]{8}=2$ |
${\text{log}}_{10}\mathrm{10,000}$ | 10 to what power is 10,000? | 4 | ${10}^{4}=\mathrm{10,000}$ |
${\text{log}}_{10}\left(\frac{1}{100}\right)$ | 10 to what power is $\frac{1}{100}$ ? | –2 | ${10}^{\mathrm{\u20132}}=\frac{1}{{10}^{2}}=\frac{1}{100}$ |
${\text{log}}_{5}0$ | 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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