Exponents and Logarithms: Rules, Proofs, and Applications

Laws of Exponents

Exponents obey three fundamental rules that all other results derive from:

• Product rule: a^m \cdot a^n = a^{m+n} — multiplying same-base powers adds exponents.
• Quotient rule: \dfrac{a^m}{a^n} = a^{m-n} — dividing subtracts exponents.
• Power rule: (a^m)^n = a^{mn} — a power of a power multiplies exponents.

These combine with the zero-exponent identity a^0 = 1 \text{ (for } a \neq 0\text{)}, which follows logically from the quotient rule applied to equal powers.

Fractional and Negative Exponents

Negative exponents mean reciprocal, not negative values:

Fractional exponents mean roots. The denominator is the root index; the numerator stays as a power:

Together, these let you rewrite any radical expression using exponents and vice versa — essential for calculus where the power rule handles all fractional and negative exponents identically.

What Is a Logarithm?

A logarithm is the inverse of an exponential function. It answers the question: "to what power must I raise the base to get this number?"

Example: \log_2(8) = 3 \text{ because } 2^3 = 8.

Two special bases:

• log(x) (no base shown) usually means base 10 — the common logarithm.
• ln(x) means base e ≈ 2.718 — the natural logarithm.

Logarithm Properties

The three logarithm rules mirror the exponent rules — because logs are the inverse of exponents:

• Product rule: \log_b(xy) = \log_b(x) + \log_b(y) — log of product = sum of logs.
• Quotient rule: \log_b\left(\dfrac{x}{y}\right) = \log_b(x) - \log_b(y) — log of quotient = difference of logs.
• Power rule: \log_b(x^n) = n \log_b(x) — the exponent pulls out in front.

Also important: log_b(1) = 0 and log_b(b) = 1 for any base b. These rules let you combine, separate, or simplify any logarithmic expression.

Solving Exponential Equations

Strategy 1 — matching bases. If you can rewrite both sides with the same base, equate exponents directly.

Example: Solve 2^x = 32. Since 32 = 2⁵, write 2^x = 2⁵, so x = 5.

Strategy 2 — take the logarithm of both sides. When bases don't match nicely, apply log to both sides and use the power rule to bring down the exponent.

Change of Base Formula

Calculators usually have only log (base 10) and ln (base e) keys. The change of base formula lets you compute logs of any base:

Example: log₇(50) = ln(50)/ln(7) ≈ 3.912/1.946 ≈ 2.011. Either log or ln works — the ratio gives the same result.

Applications: Growth and Decay

Exponential functions model any process where the rate of change is proportional to the current amount. The general continuous form:

where A₀ is the starting amount, k is the growth rate (positive = growth, negative = decay), and t is time. Applications include compound interest, population growth, radioactive decay, and cooling.

Discrete example: Bacteria doubling each hour starting at 1000: P(t) = 1000 \cdot 2^t \text{ (doubles every hour)}. Half-life, doubling time, and compound interest are all log problems — solve for t using logarithms.

Exponential growth and decay models are analyzed using derivatives to find rates of change, and integration techniques to find accumulated quantities.

Common Algebra Mistakes

Practice Problems

1. Simplify x^5 \cdot x^{-2}.
2. Simplify \dfrac{(x^3)^4}{x^5}.
3. Rewrite \sqrt[3]{x^2} using fractional exponents.
4. Evaluate \log_5(125).
5. Expand \log\left(\dfrac{x^2 y}{z}\right) as a sum/difference of logs.
6. Solve 3^x = 20.
7. A culture starts at 500 bacteria and doubles every 3 hours. How many after 12 hours?

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