Logarithms | Math Learning Guide

📋 Prerequisites

Exponents & Radicals

Logarithms are the inverse of exponentiation — they answer "what power do I raise the base to get this number?" If exponents are multiplication on steroids, logarithms are division on steroids. They turn products into sums and powers into products, which is why they show up everywhere in science and competition math.

📚 Key Concepts

What Is a Logarithm?

Definition: $ \log_b a = x $ means $ b^x = a $

$b$ = base (must be positive, not equal to 1)
$a$ = argument (must be positive)
$x$ = the exponent you need

Examples:

$ \log_2 8 = 3 $ because $ 2^3 = 8 $
$ \log_{10} 1000 = 3 $ because $ 10^3 = 1000 $
$ \log_5 1 = 0 $ because $ 5^0 = 1 $ (log of 1 is always 0)

Common log: $ \log x $ (no base written) usually means base 10: $ \log_{10} x $

Natural log: $ \ln x = \log_e x $, where $ e \approx 2.71828 $

Logarithm Rules (The Big Four)

All derived from exponent rules. For $b > 0, b \neq 1$, and $M, N > 0$:

Product Rule: $ \log_b(MN) = \log_b M + \log_b N $ — log of product = sum of logs
Quotient Rule: $ \log_b(\frac{M}{N}) = \log_b M - \log_b N $ — log of quotient = difference of logs
Power Rule: $ \log_b(M^k) = k \cdot \log_b M $ — exponent comes down as multiplier
Identity: $ \log_b b = 1 $ and $ \log_b 1 = 0 $

Also useful: $ \log_b(\frac{1}{N}) = -\log_b N $ (special case of quotient rule)

Change of Base Formula

Convert any logarithm to a different base:

\[ \log_b a = \frac{\log_c a}{\log_c b} \]

Most commonly used to convert to base 10 or base e for calculator computation:

$ \log_2 7 = \frac{\log 7}{\log 2} \approx \frac{0.8451}{0.3010} \approx 2.807 $

Solving Logarithmic Equations

Strategy 1: Rewrite in exponential form

If $ \log_b x = k $, then $ x = b^k $

Strategy 2: Combine logs, then exponentiate both sides

Use log rules to condense to a single log, then rewrite exponentially.

Always check solutions! Log arguments must be positive — extraneous solutions are common.

Solving Exponential Equations with Logs

When you can't get the same base, take log of both sides:

Example: $ 2^x = 15 $

Take log of both sides: $ \log(2^x) = \log 15 $

Power rule: $ x \cdot \log 2 = \log 15 $

Solve: $ x = \frac{\log 15}{\log 2} $

Logarithm Graphs

$ y = \log_b x $ for $b > 1$:

Passes through (1, 0) — always
Passes through (b, 1)
Domain: $ x > 0 $ (only positive inputs)
Range: all real numbers
Increasing function (rises slowly to the right)
Vertical asymptote at $x = 0$
Reflection of $ y = b^x $ over the line $ y = x $ (inverse functions)

⚠️ Common Mistake: $ \log_b(M + N) \neq \log_b M + \log_b N $. The product rule applies to multiplication inside the log, not addition. Also: log of a negative number is undefined — always verify that your solution makes the original log arguments positive.

💡 Key Insight: In competition problems, logarithms are almost always about applying the rules to simplify expressions or solve equations. When you see addition of logs with the same base, combine them into a product. When you see a coefficient in front of a log, move it up as an exponent. The goal is usually to get a single log on each side, then drop the logs (if same base) and solve the equation inside.

✏️ Example Problems

📝 Example 1 (Evaluate a logarithm)

What is the value of $ \log_4 64 $?

Definition: $ \log_4 64 = x $ means $ 4^x = 64 $

Find x:

$ 4^1 = 4 $

$ 4^2 = 16 $

$ 4^3 = 64 $ ✓

So $ x = 3 $.

Alternative: Rewrite with base 2.

$ \log_4 64 = \log_{2^2} 2^6 = \frac{6}{2} \log_2 2 = 3 $

Answer: 3

📝 Example 2 (Combine and solve)

Solve: $ \log_2 x + \log_2 (x - 2) = 3 $
What is the positive solution?

Step 1: Combine logs using product rule (same base).

\[ \log_2[x(x - 2)] = 3 \]

Step 2: Rewrite in exponential form.

\[ x(x - 2) = 2^3 = 8 \]

Step 3: Solve the quadratic.

$ x^2 - 2x = 8 $

$ x^2 - 2x - 8 = 0 $

Factor: $ (x - 4)(x + 2) = 0 $

Solutions: $ x = 4 $ or $ x = -2 $

Step 4: Check for extraneous solutions (log arguments must be positive).

$ x = 4 $: $ \log_2 4 + \log_2 2 = 2 + 1 = 3 $. ✓ Valid

$ x = -2 $: Can't take log of a negative number. ✗ Extraneous

Answer: 4

📝 Example 3 (Change of base)

Given $ \log_2 3 \approx 1.585 $ and $ \log_2 5 \approx 2.322 $,
what is the value of $ \log_2 45 $?

Step 1: Factor 45 to use known log values.

$ 45 = 9 \times 5 = 3^2 \times 5 $

Step 2: Apply log rules.

\[ \log_2 45 = \log_2(3^2 \times 5) \]

Product rule: $ = \log_2 3^2 + \log_2 5 $

Power rule: $ = 2 \cdot \log_2 3 + \log_2 5 $

Step 3: Substitute the given values.

$ = 2(1.585) + 2.322 $

$ = 3.170 + 2.322 $

$ = 5.492 $

Check: $ 2^{5.492} \approx 45 $. ✓

Answer: 5.492