The Complete Guide to Logarithms: Rules, Examples, and Applications

Logarithms are one of the most powerful tools in mathematics, yet many students find them intimidating at first. In reality, a logarithm simply answers the question: “What power must I raise this base to in order to get that number?” Once you recognise that connection, logarithms become a natural extension of exponentiation. This guide covers the definition, the key rules, worked examples, real-world applications, and the most common mistakes to avoid.

What Is a Logarithm?

The logarithm base b of a number x is the exponent y to which b must be raised to produce x. In formal notation:

\log_b(x) = y \quad \Longleftrightarrow \quad b^y = x

For example, $\log_2(8) = 3$ because $2^3 = 8$ . The base b must be a positive number not equal to 1, and the argument x must be positive.

Common Bases

Common logarithm ( $\log_{10}$ or simply $\log$ ): base 10. Used in engineering, pH scales, and decibel measurements.
Natural logarithm ( $\ln$ ): base $e \approx 2.71828$ . Central to calculus, compound interest, and growth models.
Binary logarithm ( $\log_2$ ): base 2. Fundamental in computer science and information theory.

The Four Key Logarithm Rules

These rules let you simplify, expand, or combine logarithmic expressions. They follow directly from the corresponding exponent laws.

1. Product Rule

\log_b(MN) = \log_b(M) + \log_b(N)

The logarithm of a product equals the sum of the logarithms. This is why logarithms historically turned multiplication into addition, making calculations far simpler before electronic calculators existed.

2. Quotient Rule

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

The logarithm of a quotient equals the difference of the logarithms.

3. Power Rule

\log_b(M^k) = k \cdot \log_b(M)

An exponent inside a logarithm can be brought out front as a multiplier. This rule is particularly useful when solving exponential equations.

4. Change of Base Formula

\log_b(x) = \frac{\log_a(x)}{\log_a(b)}

This lets you convert a logarithm to any base you like. Most calculators only offer $\log$ (base 10) and $\ln$ (base e), so the change of base formula is essential for evaluating expressions like $\log_5(30)$ .

Worked Example 1: Simplifying with Log Rules

Problem: Simplify $\log_3(27) + \log_3(9)$ .

Using the product rule:

\log_3(27) + \log_3(9) = \log_3(27 \times 9) = \log_3(243)

Since $3^5 = 243$ , the answer is 5.

Worked Example 2: Solving an Exponential Equation

Problem: Solve $5^x = 200$ .

Take the logarithm of both sides (using any base):

x = \log_5(200) = \frac{\ln(200)}{\ln(5)} = \frac{5.2983}{1.6094} \approx 3.292

So $x \approx 3.292$ .

Worked Example 3: Change of Base

Problem: Evaluate $\log_7(50)$ using common logarithms.

\log_7(50) = \frac{\log(50)}{\log(7)} = \frac{1.6990}{0.8451} \approx 2.010

This tells us that $7^{2.010} \approx 50$ .

Real-World Applications

pH Scale (Chemistry)

The pH of a solution measures its acidity on a logarithmic scale:

\text{pH} = -\log_{10}[\text{H}^+]

Each whole-number drop in pH represents a tenfold increase in hydrogen ion concentration. Lemon juice (pH 2) is 100 times more acidic than orange juice (pH 4).

Richter Scale (Seismology)

Earthquake magnitude is measured logarithmically. A magnitude 7 earthquake releases about 31.6 times more energy than a magnitude 6 earthquake. The formula relating energy E to magnitude M is:

\log_{10}(E) = 1.5M + 4.7

Decibels (Acoustics)

Sound intensity in decibels uses a logarithmic ratio:

L = 10 \cdot \log_{10}\!\left(\frac{I}{I_0}\right)

where $I_0$ is the reference intensity (threshold of hearing). A sound that is 10 times more intense registers as 10 dB louder.

Common Mistakes to Avoid

Writing $\log(a + b) = \log(a) + \log(b)$ . This is incorrect. The product rule applies to $\log(ab)$ , not to $\log(a + b)$ . There is no simple rule for the logarithm of a sum.
Forgetting domain restrictions. You cannot take the logarithm of zero or a negative number (in real-number mathematics). Always check that the argument is positive.
Confusing $\log(x^2)$ with $(\log x)^2$ . The first equals $2\log(x)$ by the power rule. The second is the square of $\log(x)$ , which is something entirely different.
Misapplying the quotient rule in reverse. $\log(a) - \log(b) = \log(a/b)$ , not $\log(a - b)$ .

Tips for Working with Logarithms

Memorise key values: $\log_{10}(10) = 1$ , $\ln(e) = 1$ , and $\log_b(1) = 0$ for any base.
When solving equations, isolate the logarithm first, then convert to exponential form.

Try It Yourself

Ready to practise? Use our free Logarithm Calculator to evaluate logarithms of any base, verify your manual calculations, and explore the rules covered in this guide. You can also try our Exponent Calculator to see the relationship between logarithms and powers.