We use logarithms to help us differentiate.

Before the days of calculators and computers, this was critical knowledge for anyone in a computational discipline.

Using the table again, we see that . Since we were working in scientific notation, we need to multiply this by . Our final answer is Since , this is a good approximation.

**Logarithms allow us to use addition in place of
multiplication.**

### Logarithmic differentiation

When taking derivatives, both the product rule and the quotient rule can be cumbersome to use. Logarithms will save the day. A key point is the following which follows from the chain rule. Let’s look at an illustrative example to see how this is actually used.

The process above is called *logarithmic differentiation*. Logarithmic differentiation
allows us to compute new derivatives too.

The next example will be useful when we want to use logarithmic differentiations for functions that assume negative values.

### Proof of the power rule

Finally, recall that previously we only proved the power rule for positive integer exponents. Now we’ll use logarithmic differentiation to give a proof for all real-valued exponents. We restate the power rule for convenience sake:

We will use logarithmic differentiation. Set . Write

Now differentiate both sides, and solve for

Whenever is in the domain of , the definition of the derivative at implies that , if
and , if , so the power rule applies to , too.

Thus we see that the power rule holds for all real-valued (nonzero) exponents.

While logarithmic differentiation might seem strange and new at first, with a little practice it will seem much more natural to you.