12C

The logarithmic function base e

The natural logarithm ln x as the inverse of e^x, combining ln and exp, solving equations with ln, and transformations of y = ln x.

The natural logarithm

All the log laws hold: $\ln xy = \ln x + \ln y$, $\ln \dfrac{x}{y} = \ln x - \ln y$, $\ln x^n = n\ln x$, and $\ln 1 = 0$, $\ln e = 1$.

ln x is the reflection of eˣ

Orange: y = eˣ. Blue: y = ln x, its reflection in y = x. Move P along the log curve.

Try this

Where is $\ln x$ negative?

Solving equations

Example 1●

Solve $e^x = 5$ and $\ln x = 2$, correct to 3 decimal places.

$x = \ln 5 \approx 1.609$

$x = e^2 \approx 7.389$

Example 2●●

Solve $e^{2x} = 9$ exactly.

$2x = \ln 9 = 2\ln 3$

$x = \ln 3 \approx 1.099$

Transformations of y = ln x

$y = \ln x$ passes through $(1, 0)$ with gradient $1$ there, and has vertical asymptote $x = 0$. $y = \ln(x - h) + k$ has asymptote $x = h$ and domain $x > h$. $y = \ln(-x)$ reflects in the $y$-axis and has domain $x < 0$.

Example 3●●

State the domain of $y = \ln(x - 3)$ and find its $x$-intercept.

Need $x - 3 > 0$: domain $x > 3$, asymptote $x = 3$.

$\ln(x - 3) = 0$ when $x - 3 = 1$: $x = 4$.

Flashcards

$ln x$

$\log_e x$

$e^{ln x}$

$x$

$ln e^x$

$x$

$ln 1$ and $ln e$

$0$ and $1$

Practice

$\ln x$ means:
Simplify $\ln e^3$.
Simplify $e^{\ln 7}$.
Solve $e^x = 5$, correct to 3 decimal places.
Solve $\ln x = 2$, correct to 3 decimal places.
Evaluate $\ln 1$.
The domain of $y = \ln(x - 3)$ is:
Solve $e^{2x} = 9$, correct to 3 decimal places.
Simplify $\ln\left(e^2 \times e^3\right)$.