Natural Logarithm

The natural logarithm of a number $x$ is the logarithm to the base $e$ , where $e$ is the mathematical constant approximately equal to $2.718$ . It is usually written using the shorthand notation $\ln x$ , instead of $\log_{e} x$ as you might expect . You can rewrite a natural logarithm in exponential form as follows:

$\ln x = a \Leftrightarrow e^{a} = x$

Example 1:

Find $\ln 7$ .

On a scientific calculator, you can simply press $[7]$ followed by $[\ln]$ to get the answer: approximately $1.946$ .

The exponential form of the equation you're solving is

$\begin{array}{l} e^{a} = 7 \\ {2.718}^{1.954} \approx 7 \end{array}$

Example 2:

Solve the equation. Round to the nearest thousandth.

$\ln x = - 5.5$

First, rewrite the equation in exponential form.

$e^{- 5.5} = x$

Use a calculator. (Most scientific calculators have a button which gives a good approximation for $e$ ; if yours doesn't have one, use $2.71828$ .)

$x \approx 0.004$

The usual properties of logarithms are also true for the natural logarithm.

Example 3:

Simplify.

$\ln {(3 q)}^{2}$

The following property lets you simplify logarithms of a power:

$\log_{b} x^{y} = y \log_{b} x$

So,

$\ln {(3 q)}^{2} = 2 \ln (3 q)$

Now use the property that the log of a product is equal to the sum of the logs.

$\log_{b} x y = \log_{b} x + \log_{b} y$

So,

$\begin{array}{l} 2 \ln (3 q) = 2 (\ln 3 + \ln q) \\ \approx 2.197 + 2 \ln q \end{array}$

The graph of the logarithmic function $f (x) = \ln x$ (shown in blue, below) looks similar to the graphs of related functions $g (x) = \log_{2} x$ or $h (x) = \log x$ (remember that if no base is written, the base of the logairthm is understood to be $10$ ).

The function has an asymptote at $x = 0$ and an $x$ -intercept at $(1, 0)$ . It passes through the points $(\frac{1}{e}, - 1)$ and $(e, 1)$ .

Natural Logarithm

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