Math 30-1 · Exponents and logs
Exponents and logarithms
A logarithm is just a question: "what exponent do I need?" Once that clicks, the laws stop looking like rules to memorise and start looking obvious.
Exponential and log graphs
The idea: y = bˣ and y = logb x are inverses, so their graphs are reflections of each other in the line y = x.
| y = bˣ | y = logb x | |
|---|---|---|
| Domain | all real numbers | x > 0 |
| Range | y > 0 | all real numbers |
| Intercept | (0, 1) | (1, 0) |
| Asymptote | horizontal, y = 0 | vertical, x = 0 |
When b > 1 the exponential grows; when 0 < b < 1 it decays. Transformations work the way they always do: y = log(x − 2) moves the asymptote to x = 2, so its domain is x > 2.
What a logarithm actually is
The idea: logb x = y means exactly the same thing as by = x. The log is the exponent.
Read log₂ 8 as "2 to what power gives 8?" The answer is 3. Every evaluation works this way:
- log 1000 = 3, because 10³ = 1000 (no base written means base 10)
- log₅ 1 = 0, because anything to the power 0 is 1
- log₃(1/9) = −2, because 3⁻² = 1/9
- logb b = 1, always
Watch out: you cannot take the log of zero or a negative number. No power of a positive base ever gives you one.
The laws of logarithms
The idea: logs turn multiplying into adding, dividing into subtracting, and powers into multipliers — because that is exactly what exponents do.
| Law | Rule | Example |
|---|---|---|
| Product | log(MN) = log M + log N | log 25 + log 4 = log 100 = 2 |
| Quotient | log(M/N) = log M − log N | log₂ 12 − log₂ 3 = log₂ 4 = 2 |
| Power | log(Mⁿ) = n log M | 2 log 5 = log 25 |
| Change of base | logb x = log x ÷ log b | log₃ 20 = log 20 ÷ log 3 ≈ 2.73 |
Writing logs in terms of other logs
If log 2 = a and log 3 = b, then log 12 = log(2² × 3) = 2 log 2 + log 3 = 2a + b. Break the number into factors you know, then apply the laws.
Watch out: log(M + N) is not log M + log N, and log M ÷ log N is not log(M/N). The laws only work in the shapes shown above.
Solving equations and real-world models
The idea: if you can make the bases match, match the exponents. If you cannot, take the log of both sides.
Method 1: common base
4ˣ⁺¹ = 8ˣ. Both are powers of 2: 2²⁽ˣ⁺¹⁾ = 2³ˣ. Now the exponents must be equal: 2x + 2 = 3x, so x = 2.
Method 2: take logs
3ˣ = 20 has no common base. Take the log of both sides: x log 3 = log 20, so x = log 20 ÷ log 3 ≈ 2.73. Sanity check: 3² = 9 and 3³ = 27, so the answer must be between 2 and 3.
Log equations: always check
log₂ x + log₂(x − 2) = 3 combines to x(x − 2) = 2³ = 8, which gives x = 4 or x = −2. But log₂(−2) is undefined, so −2 is extraneous. The only solution is x = 4.
The models you will see
- Growth and decay: A = A₀(b)t/p, where p is how long one doubling or halving takes. 80 g with a 5-year half-life after 15 years: 80(½)³ = 10 g.
- Compound interest: A = P(1 + i)ⁿ. To double $1000 at 6%: 2 = 1.06ⁿ, so n = log 2 ÷ log 1.06 ≈ 11.9 years.
- Log scales: Richter, pH and decibels go up by a factor of 10 per step. Magnitude 7 is 10² = 100 times as intense as magnitude 5.
Watch out: on a log scale, a difference of 2 means 100 times, not twice. The number on the scale is the exponent.