1 · IB Maths
Exponentials and logarithms
Exponential growth hides its structure on an ordinary axis. Switch to a logarithmic axis and the curve becomes a straight line, and use that line to solve a·bˣ = k.
Predict first
Start from a = 2, b = 1.5 (growth). If you raise the base b to 3.00, what happens to the 'doubles each time x rises by' value in the readout?
The doubling step is log 2 ÷ log b. A larger b has a larger log b, so the step shrinks (about 1.71 down to 0.631).
Check it in the model: Move the Base, b slider from 1.50 to 3.00 and read the Growth line in the readout.
y = a·bˣ level y = k and its solution
log y = log a + x log b (straight line)
gradient = log b, intercept = log a
solve a·bˣ = k: x = log(k ÷ a) ÷ log b
doubling or halving step: log 2 ÷ |log b|
(log means log₁₀ here)
Try this
- Set a = 2 and b = 2, then tick Logarithmic y-axis. Read the gradient of the straight line from the readout and compare it with log 2.
- Keep a = 2, b = 1.5 and move k to 100. Check the solution x = log(k ÷ a) ÷ log b against where the dashed lines meet the curve.
- Set b below 1 (try 0.7) with a = 10. Which way does the log-axis line slope, and what is the sign of log b?
Exam tip: to solve bˣ = k, take logs of both sides: x = log k ÷ log b. A straight line on a log graph of y against x means y = a·bˣ, with gradient log b and intercept log a.
More on 1: Arithmetic and geometric series · Compound interest and depreciation · Proof by induction · Permutations and combinations · De Moivre's theorem · Systems of linear equations · Complex numbers on the Argand plane · Roots of a complex number · Complex loci · Matrix transformations · Sequences and series · Binomial theorem