Exponential and logarthmic functions | Khan Academy
Exploring this relationship between them, we discuss properties of the exponential and logarithm functions, including their graphs and the rules for manipulating. equations. Then analyze both logarithmic and exponential functions and their graphs. Relationship between exponentials & logarithms: graphs. (Opens a. b) What is the relationship between the graph of y = e x and the graph b) of y = e −x? Exponential functions and logarithmic functions with base b are inverses.
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. 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.
- Relationship between exponentials & logarithms: graphs
- 4.2 - Logarithmic Functions and Their Graphs
- Exponentials & logarithms
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. What is the power I need to raise b to to get to four.Graphing exponential functions - Exponential and logarithmic functions - Algebra II - Khan Academy
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.
Introduction to Logarithmic Functions
When x is equal to 16 then y is equal to log base b of The power I need to raise b to, to get to Well we already know, if we take b squared, we get to 16, so this is equal to two.
When x is equal to 16, y is equal to two. Notice on the graph that, as the value of x increases, the value of f x also increases. This means that the function is an increasing function.
Recall that an increasing function is a one-to-one-function, and a one-to-one function has a unique inverse. The inverse of an exponential function is a logarithmic function and the inverse of a logarithmic function is an exponential function.
Notice also on the graph that as x gets larger and larger, the function value of f x is increasing more and more dramatically.
Logarithmic and exponential functions - Topics in precalculus
This is why the function is called an exponential function. If you are interested in reviewing the graphs of exponential functions, examples and problems, click on Exponential.
Once you know the shape of a logarithmic graphyou can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph. Notice that the graph of this function is located entirely in quadrants I and IV. Notice also that the graph never touches the y-axis. What does that mean? It means that the value of x domain of the function f x in the equation is always positive.
Why is this so? Recall that the equation can be rewritten as the exponential function. There is no value of f x that can cause the value of x to be negative or zero. The graph of will never cross the y-axis because x can never equal 0.
LOGARITHMIC AND EXPONENTIAL FUNCTIONS
The graph will always cross the x-axis at 1. Notice on the graph that, as x increases, the f x also increases. Notice on the graph that the increase in the value of the function is most dramatic between 0 and 1.
Notice on the graph that the function values are positive for x's that are greater than 1 and negative for x's less than 1.