Defining Logarithms

For real logarithms, the conditions are:

• $a$ is the base
• $b$ is the argument
• $c$ is the exponent returned by the logarithm

The conditions match the behavior of the exponential function. A valid exponential base is positive and not equal to one, and $a^c$ is always positive, so a real logarithm cannot have a nonpositive argument.

Converting Between the Two Forms

Each row states the same relationship twice:

When converting, keep the same base. The exponential output becomes the logarithm's argument, and the exponent becomes the logarithm's value. In the third row, a negative exponent produces a fractional output.

Common and Natural Logarithms

A logarithm with base $10$ is called the common logarithm. Its base is often omitted:

A logarithm with base $e$ is called the natural logarithm and is written $\ln x$. The constant $e\approx2.71828$ is Euler's number. Both notations are special cases of $\log_a x$.

Solving for an Unknown Exponent

Suppose an idealized culture starts with $2{,}000$ bacteria and doubles every hour:

To find when the continuous model reaches $100{,}000$, set the population equal to the target. Divide both sides by the initial population before finding the exponent:

Using a calculator or change of base:

The continuous curve crosses $100{,}000$ at about $5.64\text{ hours}$. If the culture is counted only after completed hourly divisions, the interpretation changes:

So the first completed hourly interval at or above the target is $6\text{ hours}$. The logarithm solves the equation. The context determines how that real-valued answer should be interpreted.

On the graph, the target lies between two hourly observations. The point after five hours is below the target, while the point after six hours is above it.

Published: Apr 4, 2025. Updated: Sep 1, 2026.