# Discuss the relationship between exponential and logarithmic functions

What is the relationship between exponents and logarithms? How would you To elaborate a logarithm is a inverse of an exponential function. What a function. There is one relationship you can derive about the derivatives. . and rigor I have not mentioned, but I hope this gives you a bit of a flavor of what is going on. Actually the logarithm function is defined as the inverse of exponent function. The exponential and logarithmic functions are inverse functions of each other. Exploring this relationship between them, we discuss properties of the exponential.

This point is telling us that b to the second power is equal to When x is equal to two, b to the second power, y is equal to Now we want to plot the three corresponding points on this function. Let me draw another table here.

Now it's essentially the inverse function where this is going to be x and we want to calculate y is equal to log base b of x. What are the possibilities here? What I want to do is think Let's take these values because these are essentially inverse functions log is the inverse of exponents.

### Logarithmic and exponential functions - Topics in precalculus

If we take the points one, four, and What is y going to be here? This is saying what power I need to raise b to to get to one. If we assume that b is non zero and that's a reasonable assumption because b to different powers are non zero, this is going to be zero for any non zero b. This is going to be zero right there, over here. We have the point one comma zero, so it's that point over there.

Notice this point corresponds to this point, we have essentially swapped the x's and y's. In general when you're taking an inverse you're going to reflect over the line, y is equal to x and this is clearly reflection over that line. Now let's look over here, when x is equal to four what is log base b of four.

## Relationship between exponentials & logarithms: graphs

What is the power I need to raise b to to get to four. We see right over here, b to the first power is equal to four. We already figured that out, when I take b to the first power is equal to four.

This right over here is going to be equal to one. When x is equal to four, y is equal to one. Notice once again, it is a reflection over the line y is equal to x.

Logarithms undo exponential and are the inverse of exponential. The logarithm function can be undone by the exponential function and vice versa. To distinguish between the two: An exponent of a number states the number of times to use it in multiplication. A logarithm asks what exponent produced this or how many of one number to multiply to get another number.

The logarithm gives you the exponent as the answer. And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. In order to graph a logorithm, you have to turn it into an exponential form.

Usually logarithms are written and used as bases 10 or the number e. When e is used it is called the natural log. They all follow the graphs of exponential functions and change according to the numbers they are attached to.

Logarithms do the same thing only on a grander scale. Response 6 A Logarithm says how many of one number to multiply to get another number. The exponent of a number says how many times to use the number in a multiplication. Response 7 An exponent is a number that stands for the power at which an associated number or expression is raised to.