# Subtracting logarithms with different bases of a relationship

### Evaluating logarithms (advanced) (video) | Khan Academy

A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to. You can do a bit more simplification. The important properties of the log function are, for any base a > 0,. log a (b c) = log a b + log a c, so, for example. Now that we have a good reason to pick a particular base, we will be talking a lot about the new function $ e^x $ and its inverse function $ log_e(x) $.

And I'll give you an even further hint. Now you might be saying wait, wait, wait, we wrote a logarithm here but you didn't write what the base is.

## Using the logarithm change of base rule

Well this is going to be true regardless of which base you choose as long as you pick the same base. This could be base nine, base nine in either case. Now typically, people choose base So 10 is the most typical one to use and that's because most peoples calculators or they might be logarithmic tables for base So here you're saying the exponent that I have to raise A to to get to B is equal to the exponent I have to raise 10 to to get to B, divided by the exponent I have to raise 10 to to get to A.

- Evaluating logarithms (advanced)
- Math Insight
- Evaluating logarithms (advanced)

This is a really really useful thing to know if you are dealing with logarithms. And we prove it in another video.

But now we'll see if we can apply it. So now going back to this yellow expression, this once again, is the same thing as one divided by this right over here. So let me write it that way actually. This is one divided by log base B of four. Well let's use what we just said over here to re-write it.

So this is going to be equal to, this is going to be equal to one, divided by, instead of writing it log base B of four, we could write it as log of four, and if I just, if I don't write the base there we can assume that it's base 10, log of four over log of B.

Now if I divide by some fraction, or some rational expression, it's the same thing as multiplying by the reciprocal. So this is going to be equal one times the reciprocal of this. Log of B over log of four, which of course is just going to be log of B over log of four, I just multiplied it by one, and so we can go in the other direction now, using this little tool we established at the beginning of the video.

### Logarithmic Properties

This is the same thing as log base four of B, log base four of B. So we have a pretty neat result that actually came out here, we didn't prove it for any values, although we have a pretty general B here. If I take the, If I take the reciprocal of a logarithmic expression, I essentially have swapped the bases. This is log base B, what exponent do I have to raise B to to get to four? And then here I have what exponent do I have to raise four to to get to B?

Now it might seem a little bit magical until you actually put some tangible numbers here. Then it starts to make sense, especially relative to fractional exponents. Example 4 Simplify each of the following logarithms. When we say simplify we really mean to say that we want to use as many of the logarithm properties as we can. In order to use Property 7 the whole term in the logarithm needs to be raised to the power. We do, however, have a product inside the logarithm so we can use Property 5 on this logarithm.

### Algebra - Logarithm Functions

In these cases it is almost always best to deal with the quotient before dealing with the product. Here is the first step in this part.

Therefore, we need to have a set of parenthesis there to make sure that this is taken care of correctly. The second logarithm is as simplified as we can make it. Also, we can only deal with exponents if the term as a whole is raised to the exponent. It needs to be the whole term squared, as in the first logarithm. Here is the final answer for this problem.

## Logarithmic Properties

This next set of examples is probably more important than the previous set. We will be doing this kind of logarithm work in a couple of sections. Example 5 Write each of the following as a single logarithm with a coefficient of 1.