Exponential functions model multiplicative change, while logarithms are inverse operations that solve for exponents. Use properties carefully, keep log arguments positive, and interpret parameters in context.
- Exponential growth uses repeated multiplication, not repeated addition.
- log_b(a) answers the exponent on base b needed to get a.
- Domain rules matter: log arguments must be strictly positive.
Connect exponentials and logs
The statements b^y = x and log_b(x) = y are equivalent when b > 0 and b != 1. This equivalence is the main tool for solving equations where the variable appears in an exponent.
Common and natural logs are base 10 and base e. Calculator use is easier with these bases, but change-of-base can handle any positive base not equal to 1.
Model growth and decay
An exponential model often appears as y = a(1 + r)^t for periodic growth or decay. Here a is initial amount, r is rate per period, and t is number of periods.
- Growth: r > 0.
- Decay: -1 < r < 0.
- Doubling or half-life questions often lead to logarithmic solving steps.
Solve a growth-time question
A culture starts at 500 cells and grows 12% per hour: N = 500(1.12)^t.
- Set target 1000 cells: 1000 = 500(1.12)^t, so 2 = (1.12)^t.
- Take natural log of both sides: ln(2) = t ln(1.12).
- Solve t = ln(2)/ln(1.12).
- Compute approximation t about 6.12 hours.
Common mistakes
- Applying log rules to sums, such as log(a+b) = log(a) + log(b).
- Ignoring domain and allowing nonpositive log arguments.
- Confusing growth rate r with growth factor 1 + r.
- Rounding too early and changing the final time estimate.
Try one
Rewrite log_3(81) = x in exponential form and solve.
Exponential form is 3^x = 81, so x = 4.
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.