You already read an index statement such as \(2^3=8\) as "2 raised to the power 3 gives 8". A logarithm records the same fact from the other direction: it names the power itself. The logarithm is the inverse of exponentiation.
Definition: \(a^x=N\iff\log_a N=x\), where \(a>0\), \(a\ne 1\), \(N>0\)
| Index form | Logarithmic form |
|---|---|
| \(3^4=81\) | \(\log_3 81=4\) |
| \(5^0=1\) | \(\log_5 1=0\) |
| \(10^{-2}=0.01\) | \(\lg 0.01=-2\) |
| \(2^x=7\) | \(\log_2 7=x\) |
The base of the logarithm is the base of the power; the answer of a logarithm is the index. Converting between the two forms is the single most used move in this topic.
Two special bases
| Base | Notation | Name |
|---|---|---|
| \(10\) | \(\lg x=\log_{10}x\) | Common logarithm |
| \(e\approx 2.718\) | \(\ln x=\log_e x\) | Natural logarithm |
Basic results (from the definition)
| Result | Reason |
|---|---|
| \(\log_a a=1\) | \(a^1=a\) |
| \(\log_a 1=0\) | \(a^0=1\) |
| \(\log_a a^x=x\) | The logarithm returns the index of the power. |
| \(a^{\log_a x}=x\) | Exponentiation undoes the logarithm. |
\(\log_a y\) is defined only when \(y>0\), \(a>0\) and \(a\ne 1\). A logarithm of zero or of a negative number does not exist.
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