This lesson explains what a logarithm is in simple terms: the exponent needed to raise a base to a given number. It covers the log definition, how logs and exponents are two views of the same relationship, common logs, natural logs (ln), a worked example, and everyday uses of logarithms.
What is a logarithm?
A logarithm is the exponent that a fixed number, called the base, must be raised to in order to produce a given value. In other words, a logarithm answers the question: "what power do I raise the base to, so that I get this number?"
We write this relationship as \( \log_b(x) = y \), which means exactly the same thing as \( b^y = x \). The logarithm and the exponent are two sides of the same coin, just written differently. If you already feel comfortable with exponents from exponents in the order of operations, a logarithm is simply the operation that undoes them.
Logarithm meaning: rewriting exponents backwards
Suppose you know that \( 2^3 = 8 \). The logarithm form of this same fact is \( \log_2(8) = 3 \). Both statements say the identical thing: raising 2 to the power of 3 gives 8, and the exponent that turns 2 into 8 is 3.
Because logarithmic form and exponential form describe the same relationship, you can always flip between them. A full walkthrough of that skill is covered in converting a logarithm to exponential form, which is worth reviewing once the basic idea here feels solid.
Common logarithms and natural logarithms (what is ln in math)
When no base is written, such as \( \log(x) \), the base is assumed to be 10. This is called a common logarithm.
A natural logarithm, written \( \ln(x) \), uses the special constant \( e \approx 2.718 \) as its base. So \( \ln(x) = \log_e(x) \). Natural logarithms appear constantly in science and higher math because many natural growth and decay processes are described using base \( e \).
A worked example
Evaluate \( \log_2(8) \).
Ask the defining question: 2 raised to what power gives 8? Since \( 2^3 = 8 \), the exponent is 3, so \( \log_2(8) = 3 \).
Try another: \( \log_2(1) \). Since any nonzero base raised to the power 0 equals 1, \( 2^0 = 1 \), so \( \log_2(1) = 0 \). This is why every logarithm graph crosses the x-axis at \( x = 1 \), no matter the base.
Graph of \( y = \log_2(x) \), showing the curve passing through \( (1, 0) \), \( (2, 1) \), and \( (8, 3) \).
Notice the curve only exists for \( x > 0 \): you cannot raise a positive base to any real power and get zero or a negative number, so a logarithm's input must always be positive.
What are logarithms used for?
Logarithms are the natural tool whenever a quantity changes by multiplying rather than adding, and you need to work backward to find an exponent. A few familiar examples:
pH in chemistry is a logarithmic scale measuring acidity.
Decibels measure sound intensity on a logarithmic scale.
The Richter-type magnitude scale for earthquakes is logarithmic.
Compound interest, population growth, and radioactive decay formulas all use logarithms to solve for time or rate.
Building on this definition
Once the definition of a logarithm feels solid, the next steps are learning to evaluate logarithms without a calculator by recognizing exponent patterns, and then solving logarithmic equations where the unknown sits inside the log itself.