9709 · 3.2
Logarithmic and exponential functions
Logarithms are the inverse operation of exponentiation, much like division is the inverse of multiplication. The natural logarithm, ln(x), is the specific tool used to 'undo' the natural exponential function, e^x, which is fundamental to modelling natural processes.
Need to know
What you need to know
- Inverse relationship: $e^{\ln x} = x$ for $x > 0$, and $\ln(e^x) = x$ for all $x$.
- Graph of $y=\ln x$ is the reflection of $y=e^x$ in the line $y=x$.
- Domain of $e^x$ is $(-\infty, \infty)$, Range is $(0, \infty)$.
- Domain of $\ln x$ is $(0, \infty)$, Range is $(-\infty, \infty)$.
- Remember: $\ln x$ is the same as $\log_e x$.
Explanation
Unlocking Exponentials with Logs
- y = e^x and y = ln x are inverse functions — reflection in y = x. | Sim hint: Domain of ln x is x > 0.
- Laws of logarithms: ln(ab) = ln a + ln b; ln(a^n) = n ln a. | Sim hint: Use to solve a^x = b.
- Differentiate e^{f(x)} and ln(f(x)) — chain rule. | Sim hint: d/dx e^{2x} = 2e^{2x}.
- Modelling growth/decay: N = N₀e^{kt}. | Sim hint: Half-life from k = ln 2 / t_{½}.