Logarithms

Exponential functions answer “what do I get if I multiply by , times?” Sooner or later you need the reverse question: how many times do I multiply by to reach a given number? How many years until an investment doubles, how many halvings until a drug is gone, how many powers of ten separate two earthquakes? That reverse question is a logarithm, and it is the key that unlocks every exponential equation later in this chapter.

What Is a Logarithm?

For a base , , and a positive number :

The logarithm is the exponent that must be raised to in order to produce . It is the opposite of an exponential: exponentiation takes an exponent and gives a result; a logarithm takes the result and gives back the exponent.

A few examples make the idea concrete:

Logarithmic Notation

In , each position has a name:

Read aloud as “log base of .” Two bases are so common they get shorthand: means , and means . Both are covered in the common and natural logarithms lesson.

Converting a Logarithm to Exponential Form

To change a logarithm to exponential form, keep the base as the base and make the logarithm’s value the exponent:

A reliable way to remember it is to trace a loop: start at the base, go across the equals sign to the value, and come back to the argument — ” to the is .”

Converting Exponential Form to Logarithmic Form

Going the other way, the exponent is what the logarithm equals, and the result goes inside the log:

This is the move you will use to solve equations where the unknown is an exponent. has no whole-number answer, but in logarithmic form it becomes — an exact value you can then calculate.

Example 1: Logarithmic to Exponential Form (and Back)

Fractions are no different. converts to , which checks out: and .

Evaluating Logarithms by Hand

When the argument is a recognisable power of the base, evaluate the logarithm by asking the exponent question directly:

The last two rows are true for every base. They, together with two “undo” rules, are the basic properties of logarithms:

The product, quotient and power rules that let you break logarithms apart are in the logarithm rules lesson.

The Logarithm Is the Inverse of the Exponential

The inverse properties say that and undo each other — they are inverse functions. That has two consequences:

• The graph of is the graph of reflected across the line . Every point on the exponential becomes on the logarithm.
• Domain and range swap. The exponential takes any real input and gives a positive output, so the logarithm takes a positive input and gives any real output.

Worked Example

Evaluate .

Step 1 — call the unknown value and write the logarithm:

Step 2 — convert to exponential form:

Step 3 — write both sides as powers of :

Step 4 — equal bases mean equal exponents:

Step 5 — check by converting back:

So . A base between and turns large arguments into negative logarithms — the mirror image of what a base above does.

Common Mistakes to Avoid

• Swapping the argument and the exponent. means , not or . The base always stays the base.
• Taking the log of zero or a negative number. and are undefined — no power of is zero or negative. A negative answer is fine (); a negative argument is not.
• Treating as division. is , not . A logarithm counts factors; it does not divide.
• Assuming a missing base is . In algebra, means . The natural log is written .

Practice Problems

Work each one before opening the answer.

Problem 1. Rewrite in exponential form.

Problem 2. Rewrite in logarithmic form.

Problem 3. Evaluate .

Problem 4. Evaluate .

Problem 5. Solve .

Problem 6. Solve .

Two bases dominate real calculations — and — and your calculator has a key for each. The common and natural logarithms lesson explains what and mean and how to move between them.