Logarithmic and exponential functions

The number that grows forever

• A bank account doubles every 10 years. After 100 years, it's grown by a factor of $2^{10} = 1024$.
• But what if you want to know: "After how many years will it be worth exactly 500 times the original?" That's a question only logarithms 对数 can answer.

What is a logarithm?

• A logarithm answers "what power?": if $a^x = y$ then $x = \log_a y$.
• Example: $2^x = 32 \Rightarrow x = \log_2 32 = 5$.

Earthquake strength is measured on the logarithmic Richter scale

The laws of logarithms 对数定律

• The laws of logarithms:
  • $\log(mn) = \log m + \log n$,
  • $\log\dfrac{m}{n} = \log m - \log n$,
  • $\log(m^k) = k\log m$.

Populations can grow exponentially when resources are plentiful

e and ln

• $e^x$ and the natural logarithm 自然对数 $\ln x$ are inverses: $\ln(e^x) = x$ and $e^{\ln x} = x$.
• When the unknown is in the power, take logs of both sides.

$e^x$ and $\ln x$ are inverse functions: their graphs are reflections in the line $y = x$.

Linear form 线性形式 and modelling

• Linear form: $y = Ax^n$ becomes $\ln y = \ln A + n\ln x$ — a line with gradient $n$, intercept $\ln A$.
• This lets you find the power law from experimental data by plotting $\ln y$ vs $\ln x$.
• Logarithms and indices (powers) are reverse operations.

You've got it

Write from the solution x equals 6

• Solve 2 to the power x = 64.