# 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.

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.

## 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.

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