3.11 The natural logarithm

Natural logarithms $\log_ex$loge​x 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=loge​x, 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. The log laws that we previously studied also applies to natural logarithms to help us simplify log expressions and equations.

Practice questions

Question 1

Question 2

Question 3

The graph of $y=\log_ex$y=loge​x

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 

Combining exponential and logarithmic functions

Recall from the definition of logarithms that if $y=\ln x=\log_ex$y=lnx=loge​x then $x=e^y$x=ey. 

So if we substitute $x=e^y$x=ey into $y=\ln x$y=lnx, we see that $y=\ln e^y$y=lney. In other words, raising $e$e to the power $y$y, and then taking logs on that answer restores the original $y$y. 

If we substitute $y=\ln x$y=lnx into $x=e^y$x=ey, we can also show that $x=e^{\ln x}$x=elnx. Again taking the log of $x$x, and then use that answer as the exponent that $e$e is raised to, simply restores the original $x$x.

Using the above argument, the expression $e^{\ln5.4}$eln5.4 is simply $5.4$5.4. Also, the expression $\ln e^{\sqrt{5}}$lne√5 is simply $\sqrt{5}$√5.