Exponentials & Logarithms to the Base e

Video lesson · from 35:042 worked examples

One lesson video covers all of Exponentials & Logarithms, so it opens at 35:04 for this topic — not from the beginning.

$\log_e x$ is a logarithm to the base $e$ . This is also known as the natural logarithm of $x$ , and is often written as $\ln x$ .

\ln x = \log_e x

Usual properties of logarithms can be applied to the natural logarithm function.

Without using a calculator, value of:

$\ln(e^2) = 2$
$\ln(e^5) = 5$
$\ln(\sqrt{e}) = \frac{1}{2}$
$\ln e = 1$
$\ln(3e) = \ln 3 + \ln e = \ln 3 + 1$
$\ln(13e) = \ln 13 + \ln e = \ln 13 + 1$
$\ln 0$ cannot be found because $e^x = 0$ has no real solution (the graph of $e^x$ never crosses the x-axis).

$\ln$ means $\log_e$ : it is not a different kind of operation, so every log law applies to it unchanged.
$\ln e = 1$ and $\ln 1 = 0$ : these shortcut most base- $e$ calculations.
$e^x$ and $\ln x$ undo each other: so $e^{\ln x} = x$ . This is how you clear an exponential from an equation.
Use the right calculator button: $\ln$ and $\log$ are different keys. Using $\log$ for a natural logarithm gives a wrong answer with correct-looking working.

Example 1

Use your calculator to find $\ln 8$ .

\ln 8 \approx 2.079

Example 2

Simplify $4 \ln(2e) - 3 \ln(3e)$ expressing your answer in the form $a + \ln b - \ln c$ where $a, b, c$ are whole numbers.

\begin{aligned} &= 4(\ln 2 + \ln e) - 3(\ln 3 + \ln e) \\ &= 4\ln 2 + 4(1) - 3\ln 3 - 3(1) \\ &= 4\ln 2 + 4 - 3\ln 3 - 3 \\ &= 1 + \ln(2^4) - \ln(3^3) \\ &= 1 + \ln 16 - \ln 27 \end{aligned}