### Relationship between exponentials & logarithms (practice) | Khan Academy

In this section we will discuss various methods for solving equations that involve exponential functions or logarithm functions. Then analyze both logarithmic and exponential functions and their graphs. Relationship between exponentials & logarithms: graphs. (Opens a modal) Quiz 1. Level up on the above skills and collect up to Mastery points. Start quiz. Solve various problems that focus on the relationship between a^x=b and log_a( b)=x.

We will be looking at this property in detail in a couple of sections. We will just need to be careful with these properties and make sure to use them correctly. Also, note that there are no rules on how to break up the logarithm of the sum or difference of two terms. Note that all of the properties given to this point are valid for both the common and natural logarithms.

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. 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.

### Algebra - Logarithm Functions

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. Often there will be more than one logarithm in the equation.

When this happens we will need to use one or more of the following properties to combine all the logarithms into a single logarithm. Once this has been done we can proceed as we did in the previous example.

## Using Exponential & Logarithmic Functions to Solve Finance Problems

Show Solution First get the two logarithms combined into a single logarithm. Show Solution As with the last example, first combine the logarithms into a single logarithm. It is also important to make sure that you do the checks in the original equation. Also, be careful in solving equations containing logarithms to not get locked into the idea that you will get two potential solutions and only one of these will work.

It is possible to have problems where both are solutions and where neither are solutions. There is one more problem that we should work.

This is just a quadratic equation and everyone in this class should be able to solve that. The only difference between this quadratic equation and those you are probably used to seeing is that there are numbers in it that are not integers, or at worst, fractions.

We will get these kinds of solutions on occasion. The last step to this problem is to check the two solutions to the quadratic equation in the original equation. Doing that we can see that the first solution,