Derivatives of Logs

Derivatives of logarithms summary

The derivative of ln x is 1/x and the derivative of log base a of x is 1/(x ln a).

Derivative of lnx

The derivative of lnx is
ddx(lnx)=1x.
We can prove this by using the fact that lnx and ex are inverse functions and using the derivative of ex, which is ex. Therefore we have that elnx=x. Taking the derivative of both sides of this equation (and using the chain rule on the left) we get that:
ddx(elnx)=ddx(x)elnxddx(lnx)=1xddx(lnx)=1ddx(lnx)=1x.
Which gives the derivative stated above.

Now let’s check out some problems where we use this derivative:

Example

Let h(x)=exlnx what is h(x)?

Solution

We begin by noticing that h(x) is a product of the functions ex and lnx. Therefore we can apply the product rule to get h(x)=excdotlnx+excdot1x=ex(lnx+1x).

See solution 3

Example

Let f(x)=ln(x3) what is f(x)?

Solution

There are actually two ways of doing this problem, the first of these uses the chain rule, and the second one uses log rules. Let’s take a look at them both.

For the first we begin by looking at the function as it was given to us and notice that this is a function composition with inner function g(x)=x3 and outer function h(x)=lnx, therefore when we apply the chain rule we get
f(x)=h(g(x))g(x)=1g(x)g(x)=1x33x2=3x.

Alternatively we can use log rules to rewrite
f(x)=3lnx.
In this case we can differentiate immediately using the linearity of the derivative (i.e. that the constant 3 just pulls through the derivative). This gives us:
f(x)=31x.

See solution 3

Now you try it!

Practice Problems for derivatives logarithms

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Derivative of logax

The derivative of logax is
ddx(logax)=1xlna.
This can be computed from the derivative of lnx we just computed by using logarithm rules.

Logarithm Change of Base Formula

logab=logcblogca

Proof

If we take x=logab this means that ax=b. Now let’s apply logc to both sides of the equation. This gives us
logcax=logcb.
Then using the power rule for logarithms
xlogca=logcblogab=x=logcblogca.

We using the log rule above we can rewrite f(x) as
f(x)=logax=lnxlna.
Now the 1lna is just a constant so can be pulled through the derivative leaving us with
ddx(lnxlna)=1lnaddx(lnx)=1xlna.

Example

Let g(t)=t2log3t, what is g(t)?

Solution

We can start by noticing that g(t) is a product, meaning we need to use the product rule to take the derivative. Doing this we get
g(t)=ddt(t2)log3t+t2ddt(log3t)=2tlog3t+t21tln3.

See solution 3

What are logarithms again?

The logarithm function f(x)=logax is the function that gives you the number y where ay=x. This can seem a bit complicated, but let’s look at some specific examples:

Example

What is log28?

Solution

This is just another way of asking the question: What power do we need to raise 2 in order to get 8? The answer is 3 because 23=8.

See solution 3

Example

What is log1100? Here we’re using log without a subscript to denote log10.

Solution

As in the previous question, this is another way of asking the question: What power do we need to raise 10 in order to get 1100?

In this case we know that 102=100, so 102=1100. From this we can see that log1100=2.

See solution 3

In addition to defining logab as the function that returns the exponent y that satisfies ay=b, we can also define a logarithm as the inverse of an exponential function.

Logarithms and Exponentials

The logarithmic function f(x)=logax is the inverse of the exponential function g(x)=ax. In other words:
alogax=x and loga(ax)=x.

Check for yourself if the final two identities in the box above make sense with the definition in terms of “the exponent that gives you ay=x.”

Graphs of logarithms

The relationship between logarithms and exponentials gives us a good way to look at graphs of logarithms. In particular we can find the graph of logax by reflecting the graph of ax across the line y=x. Let’s look at the example of f(x)=3x and f1(x)=log3x.

Graph of 3^x and log_3 x

Now you might be wondering why derivatives of exponentials are exponentials, but derivatives of logarithms don’t give back logarithms so nicely in the same way. Looking at the graph, and we can see immediately at least one problem, which is that the graph of the logarithm changes sign, but it’s slope is always either positive (when a>1) or negative (when a<1). Let's look again at the example of 3x and log3x, but this time also with their derivatives.

graph of 3^x, log_3 x and their derivatives

Now let’s finish off with a quick look at what happens when the base a is less than 1. Let’s look at the example log13x. For this it can be useful to remember that
(13)y=3y,
which means that log13x=log3x (since they both answer the question “what do you have to raise 1/3 to in order to get x?”). This means that the graphs of both log13x and its derivative can be gotten from the graphs of log3x and its derivative by reflecting across the x-axis.

graphs of log base 3, log base 1/3 and their derivatives

Calculus Posts

Limits and Continuity
Using Derivatives

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Integrals
Integration Techniques
Improper integrals

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