4.07 The natural logarithm
Euler's number "$e$e"
Euler's number (pronounced 'oiler') is given the symbol $e$e. It is named after the mathematician Leonhard Euler, and although you may not have seen it before, this irrational number is as famous as $\pi$π.We call the function $y=e^x$y=ex "the exponential function" to distinguish it from all other exponential functions.
Ways to approximate $e$e
There are many ways to calculate $e$e. Obviously, the button on our calculator is a great way to find $e$e. You can locate the button on the calculator to approximate values with $e$e near the "ln" button.
Here are two other ways:
- Consider the value of $\left(1+\frac{1}{n}\right)^n$(1+1n)n as n gets bigger. In fact, use your calculator to evaluate $\left(1+\frac{1}{10000}\right)^{10000}$(1+110000)10000. What do you get?
- The value of $e$e is also found by the infinite sum: $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\ldots$10!+11!+12!+13!+14!+15!+… (the symbol "!" means factorial). If you type the first six terms of the sum into the calculator it will give you $2.7166666667$2.7166666667. The more terms you add, the better the approximation.
Practice questions
Question 1
Use a calculator or other technology to approximate the following values correct to four decimal places:
$e^4$e4
$e^{-1}$e−1
$e^{\frac{1}{5}}$e15
Question 2
The population $P$P of a city increases according to the formula $P=3000e^{kt}$P=3000ekt where $t$t is measured in years and $k$k is a constant.
Find the initial population.
If the population increases to $8000$8000 in $2$2 years find the value of $k$k.
Leave your answer in exact form.
How many complete years will it take for the population to at least double?
The graph of the exponential function
The usual methods of shifting and reflecting graphs can be applied to $y=e^x$y=ex. The graph of $y=e^x$y=ex is one of the most important graphs in this course and we need to be very familiar with its shape and properties.
Properties of the exponential function:
- The domain is all real $x$x, the range is $y>0$y>0
- There are no $x$x-intercepts as the curve is always above the $x$x-axis
- The $x$x-axis ($y=0$y=0) is a horizontal asymptote to the curve
- The curve has a gradient $1$1 at it's $y$y-intercept $\left(0,1\right)$(0,1). (We can see this by substituting $x=0$x=0 into$y'=e^x$y′=ex)
- The curve is always increasing at an increasing rate and is always concave up.
Natural logarithms $\log_ex$logex or $\ln x$lnx
Natural logarithms are logarithms to the base $e.$e.We call this the "logarithmic function" to distinguish it from other logarithmic functions with bases other than $e$e.
When we rearrange $y=e^x$y=ex into logarithmic form we get the natural logarithmic function $y=\log_ex$y=logex, which is also written as "$\ln x$lnx" (short for "natural logarithm").
The "$\ln$ln" button on the calculator can help us evaluate logarithmic functions with base $e$e.
Practice questions
Question 3
Find the value of $\ln94$ln94 correct to four decimal places.
Question 4
Find the value of $\ln\left(18\times35\right)$ln(18×35) correct to four decimal places.
The graph of $y=\log_ex$y=logex
The exponential and the logarithmic functions are inverse functions which means that their graph will be a reflection of each other across the line $y=x$y=x (their $x$x and $y$y values are swapped).
Because they are reflections, the properties of the natural logarithm graph will correspond with the properties of $y=e^x$y=ex:
- The domain is $x>0$x>0, the range is all real $y$y
- The $y$y-axis ($x=0$x=0) is a vertical asymptote to the curve
- The curve has gradient $1$1 at its $x$x-intercept $\left(1,0\right)$(1,0)
- The curve is always decreasing and concave down