1
Model exponential change
Classify a formula or equally spaced table, translate a fixed percentage into a multiplier, and state the starting value, direction and asymptote.
Course · Algebra
Exponential and Logarithmic Functions
The variable moves into the exponent, and ordinary algebra stops working. A logarithm is the tool that brings it back down — defined as an inverse, with three laws that are the exponent laws read backwards.
log_b(x) = y ⟺ b^y = x
log(MN) = log M + log N log(M/N) = log M − log N
log(M^p) = p·log M the law that solves equations
A = P·e^(rt) e = 2.71828…
What this course is about
- Subject
- Algebra
- Lessons
- 12
- Format
- Interactive
Overview
What you will be able to do
By the end you can model constant-ratio change, use logarithms as inverse functions, solve both kinds of equation, and convert a logarithmic-scale difference into the factor it represents.
2
Use logarithms as inverses
Convert between b^y = x and log_b(x) = y, identify the bases hidden by log and ln, and evaluate exact values before rounding.
3
Transform logarithmic expressions
Expand or condense with the three laws, preserve the domain, and use change of base with the argument in the numerator and the original base in the denominator.
4
Solve and interpret
Choose matching powers or logarithms, test logarithmic candidates in the original domain, build a compounding model, and turn a log-scale difference into a ratio.
Lessons
12 lessons
First exponential functions and e; then the definition of the logarithm and its laws; then solving and applying.
Exponential FunctionsClassify exponential formulas and tables, then state their base, intercept and asymptote.Not marked
Growth and DecayTranslate percentage change into a positive-quantity model and distinguish growth from decay.Not marked
The Number eCompute finite compounding values, distinguish them from their limit, and use e exactly.Not marked
What a Logarithm IsConvert between exponential and logarithmic form and evaluate exact logarithms.Not marked
Logarithmic Functions and Their GraphsSketch a logarithmic function from its domain, asymptote, direction and anchor points.Not marked
The Laws of LogarithmsExpand or condense logarithmic expressions while preserving their domain.Not marked
Common and Natural LogarithmsIdentify the base behind log or ln and use the matching inverse identity.Not marked
Change of BaseEvaluate unfamiliar-base logarithms exactly or by a checked change of base.Not marked
Solving Exponential EquationsChoose matching powers or logarithms, solve, and reject impossible exponential equations.Not marked
Solving Logarithmic EquationsState the original domain, solve a logarithmic equation, and reject extraneous candidates.Not marked
Compound Interest and Continuous GrowthBuild periodic or continuous growth models from rates, periods and units.Not marked
Logarithmic ScalesConvert differences on a logarithmic scale into ratios of the underlying quantities.Not marked
Recommended background
Exponents, inverse functions, and graphing.
Practice suggestions
After each complete worked example, cover its answer and do the faded rehearsal before the quiz. The first strategic move is supplied; the remaining algebra, interpretation and check are yours.
Say the definition out loud until it is automatic: a logarithm is an exponent. Almost every error in this course is a step taken without that sentence in mind.
Check the domain of every solution to a logarithmic equation. log(x - 5) has nothing to say about x = 2, and the algebra will happily produce it.
Use the log-scale lesson on numbers you know. The distance from 1 to 10 being the same as 10 to 100 is either obvious or wrong-feeling, and it is worth being the first.
What it does not cover
The derivative of e^x and the reason e is the natural base for calculus. This course gives e by compounding, which is honest and complete for algebra.
Logarithms of negative and complex numbers.
Curve fitting and regression on transformed data, which is a statistics topic.