Why is there sometimes no base written?

When you see $\log(x)$ without a little number at the bottom, it is called the common logarithm, which has a base of 10. It is a shorthand because base-10 is used so frequently in science and engineering.

The notation 'ln' stands for 'natural logarithm'. It is a logarithm where the base is the mathematical constant $e$ (approximately 2.718). It is written as $\ln(x)$ instead of $\log_e(x)$ and is standard in calculus and physics.

Can a logarithm result in a negative number?

Yes, the result of a logarithm can be negative. This happens when the number you are evaluating is between 0 and 1. For example, $\log_{10}(0.1) = -1$ because $10^{-1} = 0.1$.

What is a logarithm, really?

A logarithm is simply the inverse of an exponent. When you look at an exponent, you ask, 'If I multiply a base by itself a certain number of times, what do I get?' With a logarithm, you work backwards: 'How many times do I multiply this base by itself to reach a specific number?' Think of it like taking a road trip. Exponents tell you where you end up if you drive at a certain speed for a certain time. Logarithms tell you how long you had to drive to reach your destination. They are two sides of the very same mathematical coin.

The Core Equation

Every logarithm can be translated into an exponential equation. The notation $\log_b(x) = y$ is mathematically identical to $b^y = x$ . Here, $b$ is the base, $y$ is the exponent, and $x$ is the result. If you can read one, you can read the other.

Why We Need Logarithms

Logarithms are incredibly useful when the unknown variable in your equation is trapped in the exponent. If you want to know how long it takes for a population of bacteria to triple, or for a bank account to double with compound interest, you need a way to solve for time. Logarithms pull that variable down to where you can work with it.

Where Students Slip Up

The most common mistake is forgetting that a logarithm itself represents an exponent. When you add two logarithms with the same base, you are really applying the exponent rule for multiplying numbers. Another trap is trying to take the logarithm of a negative number or zero. Since raising a positive base to any power always yields a positive result, $\log_b(-5)$ is undefined in real numbers.

Worked through

Evaluate $\log_3(81)$ without using a calculator.

First, set the expression equal to a variable: $\log_3(81) = x$ . Next, rewrite this logarithmic equation in its exponential form. The base is 3, the exponent is $x$ , and the result is 81, giving us $3^x = 81$ . Now, ask yourself: 'How many times do I multiply 3 by itself to get 81?' Since $3 \times 3 = 9$ , $9 \times 3 = 27$ , and $27 \times 3 = 81$ , we know that $3^4 = 81$ . Therefore, $x = 4$ . The logarithm $\log_3(81)$ evaluates exactly to 4.

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Where this comes from: OpenStax College Algebra: Exponential and Logarithmic Functions · Khan Academy: Algebra 2 - Logarithms unit