What are logarithms?
The logarithm is a function that is calculated with respect to a base value. The most common base values are and (Euler’s number — more later). For example, the base 10 (or common) logarithm of a given value typically denoted returns a value, such that
So, when increases by one, increases by a factor of Since the common logarithm uses 10 as its base, the inverse logarithm is
When the logarithm is calculated using the base this is referred to as the natural logarithm, and for a given value would be denoted It would return a value such that
in this case, increase by one will increase by a factor of Since the natural logarithm uses as its base, the inverse logarithm is
Most calculators have [ log ] and [ ln ] buttons for this purpose. The inverse logarithm buttons are labelled [ ] and [ ].
What are they used for?
Since logarithms (or logs) represent exponents, then exponent rules apply to logs. That is, when you multiply two numbers, you can just add the exponents, like so
When raising to a power, multiply the exponent by the power, like so
- Say I needed to calculate 13 raised to the power 4.7. From the above explanation, I can calculate the log of 13, multiply by 4.7, and take the inverse log. Here are the steps:
Now calculate the inverse logarithm (calculator: enter the number and push the [ ] button)
- What if you needed to calculate the cube root of 42, but your calculator doesn’t have a “cube root” button? Logarithms will help you! Remember that the square root is the same a raising to the 1/2 power, and taking a cube root is the same as raising to the 1/3 power, so
then finish with the inverse logarithm (use the [ ] button)
Calculus with Logarithms
At this point, go back and review Chapters 2 & 3 and refresh yourself on the “recipes” for derivatives and integrals of polynomials. Go ahead — I’ll wait.
Recall that for integrals, the “recipe” is: increase the exponent by 1, then divide by the new power. Here is part of the table from Chapter 3. The first few
powers of and their integrals are shown
But there’s a problem, a “hole” in the pattern. To see it, we need to expand the table to negative powers. But first, remember that a negative power is the
same as a positive power in the denominator. For example
So now I add a few rows to the top of the above table, like so
Here we see that the “recipe” fails for because we get division by zero.
So what is the integral of ? Turns out, it is the function we have been studying. So the natural logarithm function, fills in the “hole” in our table. Here’s what it looks like now:
And our newly learned answer looks like this when written out
Since a derivative is the opposite of the integral, we also know that
the derivative of is as you would expect. This also fills a “hole” in the derivative table; I’ll let you look at it (Chapter 2) and figure out how this fits in that table.
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