Exponent and Logarithm Rules With Worked Problems
October 21, 2026
Exponents describe repeated multiplication, and logarithms are their inverses: they answer the question 'what exponent produces this number?' Together they appear in compound interest, population growth, the Richter scale, decibels, pH, and computing. The rules look like a long list but follow from one idea, that exponents count how many times a base is multiplied.
Here are the rules with worked examples, followed by real applications that you can verify on a scientific calculator.
Skip the math and get your answer instantly:
Open the Scientific Calculator →The Laws of Exponents
Product rule: a^m × a^n = a^(m+n). Example: 2³ × 2⁴ = 2⁷ = 128. Quotient rule: a^m ÷ a^n = a^(m−n). Example: 5⁶ ÷ 5² = 5⁴ = 625. Power rule: (a^m)^n = a^(mn). Example: (3²)³ = 3⁶ = 729.
Zero exponent: a⁰ = 1 for any non-zero a. Negative exponent: a^(−n) = 1 ÷ a^n, so 10⁻³ = 0.001. Fractional exponent: a^(1/n) is the nth root, so 27^(1/3) = 3 and 16^(3/4) = (16^(1/4))³ = 2³ = 8.
The Laws of Logarithms
The logarithm base b of x, written log_b(x), is the exponent y such that b^y = x. For example, log₂(8) = 3 because 2³ = 8, and log₁₀(1000) = 3. The product rule becomes addition: log(xy) = log x + log y. The quotient rule becomes subtraction: log(x/y) = log x − log y. The power rule: log(x^k) = k × log x.
Change of base: log_b(x) = ln(x) ÷ ln(b). This allows any logarithm to be computed on a calculator that has only natural log (ln, base e ≈ 2.71828) or common log (log₁₀).
| Rule | Exponent form | Log form | Example |
|---|---|---|---|
| Product | a^m × a^n = a^(m+n) | log(xy) = log x + log y | log(20) = log 2 + log 10 |
| Quotient | a^m ÷ a^n = a^(m−n) | log(x/y) = log x − log y | log(5) = log 10 − log 2 |
| Power | (a^m)^n = a^(mn) | log(x^k) = k log x | log(2^10) = 10 log 2 |
Solving 2^x = 1000
Take the logarithm of both sides: x × log 2 = log 1000. So x = log 1000 ÷ log 2 = 3 ÷ 0.30103 = 9.966. This tells us that 2^10 = 1,024 is just above 1,000, consistent with the result. This is the same calculation behind the idea that 'ten doublings is about a thousandfold'.
Another example: 5^x = 3,125 gives x = ln 3125 ÷ ln 5 = 5, because 5⁵ = 3,125. A calculator confirms that the exact answer is 5.
Application: Doubling Time
With compound growth at rate r per period, money doubles when (1 + r)^t = 2. Taking logs: t = ln 2 ÷ ln(1 + r). At 7% growth, t = 0.6931 ÷ 0.06766 = 10.24 periods. The rule of 72 approximates this as 72 ÷ 7 = 10.3.
To find the time for an investment to grow tenfold at 7%: t = ln 10 ÷ ln 1.07 = 2.3026 ÷ 0.06766 = 34.0 years.
Application: Logarithmic Scales
Many scales are logarithmic so that huge ranges fit on a readable axis. The pH scale is −log₁₀ of hydrogen ion concentration, so each pH unit is a tenfold change. A decibel is 10 × log₁₀ of a power ratio, so an increase of 10 dB means ten times the power, and 3 dB roughly doubles it (10 × log₁₀ 2 = 3.01). The Richter scale is also logarithmic: each whole number increase corresponds to roughly 10 times the measured amplitude.
Interpreting these scales correctly prevents mistakes: a pH 4 solution is 100 times more acidic than pH 6, not twice as acidic.
Common Errors
Writing log(x + y) = log x + log y (it is false), confusing (a + b)^2 with a² + b², cancelling exponents incorrectly across addition, and forgetting that logs of zero and negative numbers are undefined for real numbers. Check your answers by substituting them back into the original equation.
Frequently Asked Questions
What is a logarithm?
The exponent to which a base must be raised to produce a given number.
What is the difference between log and ln?
Log usually means base 10 (or sometimes base 2 or e depending on context), while ln is the natural logarithm with base e.
How do I solve for an exponent?
Take the logarithm of both sides and divide: x = log(target) ÷ log(base).
Why is any number to the power of zero equal to 1?
Because a^n ÷ a^n = a^0 must equal 1 by the quotient rule.
Can you take the log of a negative number?
Not within real numbers. Logs are defined only for positive arguments.