Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Logarithmic functions. A logarithmic function is the inverse of an exponential function. So if we start with an exponential function F of X equal B to the XF of X is really just like our Y. So now we have Y equal B to the X. If we thought about then switching the X&Y, which is what we do for an inverse, switch the X&Y. Now the Y is up in the exponent and we don't know how to solve for things that are up in the exponent. Hence there's been a new type of writing of this equation where we solve for the exponent. That's called a logarithm. So if X equal B to the Y, that really is going to solve for the exponent Y equal log the base B that the exponent was on. It's a little subscript, so it's written lower of whatever the B to the Y had equaled X. So Y equal log base B of X. Inverse notation means that we use F little -1 of X little -1 up in the exponent. So if F of X equal B to the X, then it's inverse is F inverse of X equal log base B to the X. So just looking at the inverse inverse portion, X equal B to the Y is the same as Y equal log base B of X. The logarithm is meant to allow us to rewrite an equation so that we've solved for the unknown exponent. Let's look at an example. If we have log base two of eight equaling X, that's really the same thing as saying 2 to the X power equaled 8. Well, two to what power equals 8? We could think of eight as 2 * 2 * 2. Once we can get the bases the same, then we would just set the exponents equal. So X in this case is really 3. We can go from exponential form into logarithmic form. If I have 8:00 to the Y equal 100, solving for the exponent Y equal log base 8 of 100. And some calculators nowadays will actually let you type this right into your calculator to get an answer. Some do not, and an equivalent is log 100 base 10 over log base 10 of the number 8. Graphing of Y equal log base 2 of X. Well, if we know that Y equal log base 2 of X, we really know that two to the Y power equal X because the logarithm is just an exponent. So here if I make AT chart, instead of putting in values for the X, we're going to put in the values for the Y. So what is 2 ^2? What is 2 to the first? What is 2 to the zero? Anything to the zero is 1. What's 2 to the -1? That negative in the exponent means that we take the reciprocal. We get 1/2. What is 2 to the -2? That negative in the reciprocal says it's 1 / 2 ^2 or 1/4. When we graph this out at 4:00, we're going to have two and two is going to be 1 and one is 01. Half is -1 and 1/4 is -2. Well, what would happen is we put in bigger and bigger and bigger negative values. Say -10, we get 2 to the -10 or 1 / 2 to the 10th. So we're getting really, really, really small numbers, but they're positive. So here we're going to have an asymptote, but this time the asymptote is going to be along the Y axis or the line X = 0. Thank you and have a wonderful day.