5. Derivative of the Logarithmic Function

by M. Bourne

First, let's look at a graph of the log function with base e, that is:

f(x) = loge(x) (usually written "ln x").

The tangent at x = 2 is included on the graph.

Natural log - ln graph

The slope of the tangent of y = ln x at `x = 2` is `1/2`. (We can observe this from the graph, by looking at the ratio rise/run).

If y = ln x,

`x` 1 2 3 4 5
slope of graph `1` `1/2` `1/3` `1/4` `1/5`
`1/x` `1` `1/2` `1/3` `1/4` `1/5`

We see that the slope of the graph for each value of x is equal to `1/x`. This works for any positive value of x (we cannot have the logarithm of a negative number, of course).

If we did many more examples, we could conclude that the derivative of the logarithm function y = ln x is

`dy/dx = 1/x`

Note 1: Actually, this result comes from first principles.

Note 2: We are using logarithms with base e. If you need a reminder about log functions, check out Log base e from before.

Continues below

Derivative of the Logarithm Function y = ln x

The derivative of the logarithmic function y = ln x is given by:

`d/(dx)(ln\ x)=1/x`

You will see it written in a few other ways as well. The following are equivalent:

`d/(dx)log_ex=1/x`

If y = ln x, then `(dy)/(dx)=1/x`

We now show where the formula for the derivative of `log_e x` comes from, using first principles.

Tip

For some problems, we can use the logarithm laws to simplify our log expression before differentiating it.

Example 1

Find the derivative of

y = ln 2x

Example 2

Find the derivative of

y = ln x2

Derivative of y = ln u (where u is a function of x)

Unfortunately, we can only use the logarithm laws to help us in a limited number of logarithm differentiation question types.

Most often, we need to find the derivative of a logarithm of some function of x. For example, we may need to find the derivative of y = 2 ln (3x2 − 1).

We need the following formula to solve such problems.

If

y = ln u

and u is some function of x, then:

`(dy)/(dx)=(u’)/u`

where u' is the derivative of u

Another way to write this is

`(dy)/(dx)=1/u(du)/(dx)`

You might also see the following form. It means the same thing.

If

y = ln f(x),

then the derivative of y is given by:

`(dy)/(dx)=(f’(x))/(f(x)`

Example 3

Find the derivative of

y = 2 ln (3x2 − 1).

Example 4

Find the derivative of

y = ln(1 − 2x)3.

Example 5

Find the derivative of `y=ln[(sin\ 2x)(sqrt(x^2+1))]`

Differentiating Logarithmic Functions with Bases other than e

If

u = f(x) is a function of x,

and

y = logb u is a logarithm with base b,

then we can obtain the derivative of the logarithm function with base b using:

`(dy)/(dx)=(log_be)(u')/u`

where

`u'` is the derivative of u

logbe is a constant. See change of base rule to see how to work out such constants on your calculator.)

Note 1: This formula is derived from first principles.

Note 2: If we choose e as the base, then the derivative of ln u, where u is a function of x, simply gives us our formula above:

`(dy)/(dx)=(u')/u`

[Recall that logee = 1. ]

[See the chapter on Exponential and Logarithmic Functions base e if you need a refresher on all this.]

Example 6

Find the derivative of y = log26x.

Example 7

Find the derivative of y = 3 log7(x2 + 1).

Note: Where possible, always use the properties of logarithms to simplify the process of obtaining the derivatives.

Exercises

1. Find the derivative of 

y = ln(2x3x)2.

2. Find the derivative of 

y = ln(cos x2).

3. Find the derivative of 

y = x ln3 x.

4. Find the derivative of

3 ln xy + sin y = x2.

5. Find the derivative of

y = (sin x)x

by first taking logarithms of each side of the equation.