Lesson 4 of 9, 11 min read
Logarithms
The definition of a logarithm, converting between logarithmic and exponential form, evaluating logarithms, common and natural logs, and domain restrictions.
Asking for the exponent
\(5^3 = 125\). The question "what power of \(5\) gives \(125\)?" has the answer \(3\), and that answer is called the logarithm base 5 of 125.
Since \(b^y\) is always positive, you can only take the logarithm of a positive number. \(\log_2(-8)\) and \(\log_5 0\) are undefined.
Converting forms
| Logarithmic form | Exponential form |
|---|---|
| \(\log_4 64 = 3\) | \(4^3 = 64\) |
| \(\log_2 \frac{1}{16} = -4\) | \(2^{-4} = \frac{1}{16}\) |
| \(\log_{27} 3 = \frac{1}{3}\) | \(27^{1/3} = 3\) |
Evaluating logarithms
To evaluate \(\log_b x\), write \(x\) as a power of \(b\).
The last two say that \(\log_b\) and "raise \(b\) to the power" undo each other: they're inverse operations.
Common and natural logarithms
- The common logarithm has base \(10\) and is written \(\log x\) (no base).
- The natural logarithm has base \(e\) and is written \(\ln x\).
Calculators have a LOG key and an LN key. For example, \(\log 0.001 = -3\) and \(\ln\frac{1}{e} = -1\).
Solving simple log equations by converting
Domain
Practice
Quick check
What is \(\log 1000\)?
The common log has base \(10\), and \(10^3 = 1000\).
Quick check
Which of these is undefined?
You can't take the logarithm of a negative number.
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