Logarithms for Beginners: Rules, Examples and Common Mistakes

Logarithms often look intimidating at first, but they are simply another way of talking about exponents. Once you understand that link, the rules of logarithms make complete sense and become easy to remember. This guide explains what a logarithm is, the key rules, and how to use them in solved examples of the kind you will meet in Matric, FSc, O/A Level and beyond.

What Is a Logarithm?

A logarithm answers the question: "To what power must I raise the base to get this number?"

$$ \log_a x = y \quad \Longleftrightarrow \quad a^y = x $$

Here \(a\) is the base, with \(a > 0\) and \(a \neq 1\), and \(x > 0\).

Examples:

• \(\log_2 8 = 3\), because \(2^3 = 8\)
• \(\log_{10} 1000 = 3\), because \(10^3 = 1000\)
• \(\log_5 \frac{1}{25} = -2\), because \(5^{-2} = \frac{1}{25}\)

Common and Natural Logarithms

• Common log: base 10, usually written \(\log x\).
• Natural log: base \(e \approx 2.718\), written \(\ln x\).

Two Basic Facts

For any valid base \(a\):

$$ \log_a 1 = 0 \qquad \text{and} \qquad \log_a a = 1 $$

These come directly from \(a^0 = 1\) and \(a^1 = a\).

The Main Rules of Logarithms

Each rule mirrors a law of exponents.

1. Product Rule

$$ \log_a (mn) = \log_a m + \log_a n $$

Example: \(\log_2 32 = \log_2 (4 \times 8) = 2 + 3 = 5\).

2. Quotient Rule

$$ \log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n $$

Example: \(\log_3 \frac{81}{9} = 4 - 2 = 2\), and indeed \(\log_3 9 = 2\).

3. Power Rule

$$ \log_a (m^k) = k \log_a m $$

Example: \(\log_{10} 10^5 = 5 \log_{10} 10 = 5\).

4. Change of Base Rule

$$ \log_a x = \frac{\log_b x}{\log_b a} $$

This lets you evaluate any logarithm on a calculator using \(\log\) or \(\ln\).

Example: \(\log_2 10 = \frac{\log 10}{\log 2} = \frac{1}{0.30103} \approx 3.322\).

Worked Example 1: Simplifying an Expression

Simplify \(\log 8 + \log 25 - \log 2\) (base 10).

Step 1: Use the product rule on the first two terms:

$$ \log 8 + \log 25 = \log (8 \times 25) = \log 200 $$

Step 2: Use the quotient rule:

$$ \log 200 - \log 2 = \log \frac{200}{2} = \log 100 $$

Step 3: Evaluate:

$$ \log 100 = 2 \quad \text{since } 10^2 = 100 $$

Answer: \(2\)

Worked Example 2: Expanding a Logarithm

Expand \(\log_a \left(\dfrac{x^3 \sqrt{y}}{z^2}\right)\).

Step 1: Quotient rule:

$$ \log_a (x^3 \sqrt{y}) - \log_a (z^2) $$

Step 2: Product rule on the first part:

$$ \log_a x^3 + \log_a y^{\frac{1}{2}} - \log_a z^2 $$

Step 3: Power rule on each term:

$$ 3\log_a x + \frac{1}{2}\log_a y - 2\log_a z $$

Worked Example 3: Solving a Log Equation

Solve \(\log_2 (x + 3) + \log_2 (x - 3) = 4\).

Step 1: Combine using the product rule:

$$ \log_2 \left[(x + 3)(x - 3)\right] = 4 $$

Step 2: Convert to exponential form:

$$ (x + 3)(x - 3) = 2^4 = 16 $$

Step 3: Expand and solve:

$$ x^2 - 9 = 16 \quad \Rightarrow \quad x^2 = 25 \quad \Rightarrow \quad x = \pm 5 $$

Step 4: Check the domain. Logs need positive arguments. For \(x = -5\), \(x - 3 = -8 < 0\), so reject it. For \(x = 5\), \(x + 3 = 8\) and \(x - 3 = 2\), both positive.

Answer: \(x = 5\). Check: \(\log_2 8 + \log_2 2 = 3 + 1 = 4\). Correct.

Worked Example 4: Solving an Exponential Equation with Logs

Solve \(5^x = 20\), giving your answer to 3 decimal places.

Step 1: Take the common log of both sides:

$$ \log 5^x = \log 20 $$

Step 2: Apply the power rule:

$$ x \log 5 = \log 20 $$

Step 3: Divide:

$$ x = \frac{\log 20}{\log 5} = \frac{1.30103}{0.69897} \approx 1.861 $$

Check: \(5^{1.861} \approx 20.0\). Correct.

Common Mistakes

• Splitting a log of a sum. \(\log(m + n) \neq \log m + \log n\). The product rule works only for multiplication inside the log.
• Dividing logs incorrectly. \(\frac{\log m}{\log n} \neq \log m - \log n\). The quotient rule applies to \(\log \frac{m}{n}\).
• Misplacing the power. \(\log x^2 = 2\log x\), but \((\log x)^2\) is something different.
• Ignoring the domain. You cannot take the log of zero or a negative number, so always check your solutions.
• Mixing bases. The rules only combine logs with the same base. Use change of base first if needed.

Key Takeaways

• \(\log_a x = y\) means exactly the same as \(a^y = x\).
• Product, quotient and power rules turn multiplication, division and powers into addition, subtraction and multiplication.
• Use the change of base rule to evaluate any log on a calculator.
• Always check that log arguments stay positive in your final answers.

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