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.
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.
Differentiating both sides, we find Now we can solve for ,
Alternative way: We can write Therefore, by the Chain Rule and the Product Rule.
Now, it follows that
The next example will be useful when we want to use logarithmic differentiations for functions that assume negative values.
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.