Logarithmic and exponential functions
Logarithms
- A logarithm answers "what power?": if $a^x = y$ then $x = \log_a y$.
- The laws of logarithms:
- $\log(mn) = \log m + \log n$,
- $\log\dfrac{m}{n} = \log m - \log n$,
- $\log(m^k) = k\log m$.
Practice
Evaluate log₂(8).
2³ = 8, so log₂(8) = 3.
Practice
Solve 2ˣ = 8 for x.
2³ = 8, so x = 3 (or x = log₂8 = 3).
Practice
Which is a correct law of logarithms?
log(mn) = log m + log n; and log(m^k) = k log m.
e and ln
- $e^x$ and the natural logarithm $\ln x$ are inverses: $\ln(e^x) = x$ and $e^{\ln x} = x$.
- When the unknown is in the power, take logs of both sides.
- Linear form: $y = Ax^n$ becomes $\ln y = \ln A + n\ln x$ — a line with gradient $n$, intercept $\ln A$.
Practice
What is ln(e⁵)?
ln and e are inverses, so ln(e⁵) = 5.
You've got it
Key idea
- a log answers "what power?"; laws: $\log mn = \log m + \log n$, $\log m^k = k\log m$
- $e^x$ and $\ln x$ are inverses; take logs when the unknown is a power
- linear form: plot $\ln y$ vs $\ln x$ → gradient $n$, intercept $\ln A$