Logarithms & logarithmic functions
The idea
A logarithm answers one question: what exponent turns the base into this number? Writing log₂ 32 = 5 states exactly that 2⁵ = 32 — the logarithm and the exponential are the same fact read in opposite directions. You already solve equations like 3^x = 81 by spotting the exponent; logarithms make that move available even when nothing is spottable, as in 1.05^t = 1.5. They also compress huge ranges into small ones, which is why earthquake magnitude, pH, and decibels are all logarithmic scales.
Three laws carry the subject, each a translated exponent rule: log(xy) = log x + log y, log(x/y) = log x − log y, and log(x^n) = n log x. The power law is the workhorse — it pulls an unknown exponent down to ground level where ordinary algebra can reach it. The misconception to dodge is splitting log(x + y) into log x + log y; logs convert multiplication into addition, not addition into anything, and log(x + y) simply does not simplify. Remember the domain too: only positive numbers have logarithms.
Worked example
An investment of $2000 grows by 5% per year, so its value after t years is 2000 × 1.05^t. How long until the investment is worth $3000? Give t to two decimal places.
- Isolate the exponential, the only part containing t: divide both sides of 2000 × 1.05^t = 3000 by 2000 to get 1.05^t = 1.5.
- Take a logarithm of both sides (any base works; base 10 here): log(1.05^t) = log 1.5, and the power law turns the left side into t × log 1.05.
- Solve for t: t = log 1.5 ÷ log 1.05 ≈ 0.17609 ÷ 0.02119 ≈ 8.31 years.
- Sanity-check by bracketing: 1.05⁸ ≈ 1.477 falls just short of 1.5, while 1.05⁹ ≈ 1.551 overshoots, so a crossing time between 8 and 9 years — close to the low end — fits perfectly.
Answer. It takes about t ≈ 8.31 years; the investment passes $3000 during its ninth year.
Check your understanding
- How does the power rule log(x^n) = n log x follow from the exponent rule for a power of a power?
- How would you explain to a friend why log(xy) splits into a sum while log(x + y) refuses to split at all?
- What happens to log x as x slides toward zero from above, and why does no real logarithm exist for negative numbers?
- Why does the choice of logarithm base not affect the answer when you solve 1.05^t = 1.5?
Build the foundations first
Logarithms & logarithmic functions builds on these concepts. If any feel shaky, start there.