Logarithm definition is one of 3 logarithms formulas in the algebra section of this library, and it is used at high school level.
Why logarithm definition works
A logarithm answers the question what exponent. The two statements are the same fact written from opposite ends: one names the exponent, the other names the result. That is why every logarithm rule turns out to be an exponent rule wearing different clothes.
What each symbol means
$b$ is the base, $x$ the positive argument, and $y$ the exponent.
Logarithm definition: when it holds
$b>0$, $b\ne1$, and $x>0$.
When it stops applying
There is no real logarithm of zero or of a negative number, because a positive base raised to any real power stays positive. Asking for log base 2 of -8 has no answer, and log base 2 of 0 does not exist either, though the values fall without bound as the argument approaches zero.
Logarithm definition: a worked example
$\log_2 32=5$ because $2^5=32$.
The mistake to avoid
What people do: Treating the log as a division and computing log base 2 of 8 as 4.
Why it goes wrong: The bar in the notation is not a fraction. The real question is what power of 2 makes 8, and the answer is 3, because 2 times 2 times 2 is 8.
Do this instead: Say the sentence out loud: base to the what gives the argument. Then write the exponential form b to the y equals x and solve that instead.
Logarithm definition: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $b$ is the base, $x$ the positive argument, and $y$ the exponent.
- Check the conditions before substituting. $b>0$, $b\ne1$, and $x>0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Logarithms
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where logarithm definition comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about logarithm definition
Why is a base of 1 not allowed?
Because 1 to any power is always 1. That means the question has no answer for other arguments and infinitely many answers when the argument is 1, so no single value can be assigned.
What base is meant when none is written?
In most school and calculator contexts a bare log means base 10, while ln always means base e. Higher mathematics and many programming languages use a bare log for base e, so check the source.
Can a logarithm be negative?
Yes, whenever the argument is between 0 and 1. Since 2 to the -3 is one eighth, log base 2 of one eighth is -3, and the argument itself is still positive.
Is log of the base always 1?
Yes, because the base to the first power is the base. In the same way the log of 1 is always 0, since anything to the zero power equals 1.