Today we're going to take a look at a strategy for logarithm multiplication. First, let's take a look at a problem.
(Note: all variables in this post are assumed to be in their respective domains)
What is One of the first things you should notice about this is that and are powers of the bases of the logarithms, but they are paired with the wrong base. However, you might be tempted to say that but you would first need to prove that this is true:
Let's take a look at what change of base is. Change of base is a technique often used in approximating values of logarithms. Most scientific calculators cannot find logarithms with bases that aren't or but all of them have at least a button that finds the natural logarithm of a number. Change of base lets you split a logarithm so it is a lot easier to find on a calculator. Here's the formula.
If you take the logarithm base of a number then you can take the logarithm base of and divide it by the logarithm base of to find the logarithm base of For example, if you are told to find using only a scientific calculator, then change of base allows you to do this.
(Note: approximation signs are used because of rounding to decimal places.)
So now let's take a look back at the generalization. We are trying to prove that Use change of base to split the logarithms.
Using the communative property of multiplication, you get this.
Finally, using reverse change of base, you can do this.
So that's it! We have proved that Now we can go back to the original problem.
To find you can switch the bases and say that
This idea can also be applied over multiple multiplications too. Take a look at this problem.
This is equal to Using the base-switching strategy, you can rearrange this to Everything to the right of is equal to so the value is equal to
Prove that does not exist.
Prove that is divergent.
Find the value of
So you learned about multiplying logarithms. I hope that this strategy helps you in the future. Check the tag #TrevorsTips occasionally to see more problem solving strategies from me. Thanks for reading this post!