Relationship between exponentials & logarithms: graphs (video) | Khan Academy
GRAPHS OF EXPONENTIAL FUNCTIONS. By Nancy Marcus. In this section we will illustrate, interpret, and discuss the graphs of exponential and logarithmic. c) Sketch the graph of y = e −x. Graph of exponential. Inverse relations. Exponential functions and logarithmic functions with base b are inverses. The functions. The graphs of exponential and logarithmic functions with examples and applications. Includes exponential growth and decay.
Logarithm - Wikipedia
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. 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.
Relationship between exponentials & logarithms: graphs
For purposes of discussion, we will refer to the independent variable as x and the dependent variable as y or f x. In reality, you could label them anything: Remember that when we talk about the function, the function value, the value of the function, y or f xwe are talking about the value and behavior of the y part of the point x, y in the full set of the points that form the graph.
Once you know the shape of an exponential graphyou can shift it vertically or horizontally, stretch it, shrink it, reflect it, check answers with it, and most important interpret the graph. The function is always positive. There is simply no value of x that will cause the value of to be negative.
What does this mean in terms of a graph? It means that the entire graph of the function is located in quadrants I and II. Notice that the graph never crosses the x-axis. Why is that so? It is because there is no value of x that will cause the value of f x in the formula to equal 0. Notice that the graph crosses the y-axis at 1. The value of x is always zero on the y-axis. In this section we will learn how to plot quantities on a new type of graph called a logarithmic graph.
We will see how this can make it easier to determine the functional relationship between certain quantities.
2. Graphs of Exponential `y=b^x`, and Logarithmic `y=log_b x` Functions
We will motivate logarithmic graphs by giving two examples. Let t represent time and let i represent the electric current flowing in some electric circuit. The data shown in the table was collected for t and i. For later use, the natural logarithm of the entries in column 2 was taken and entered in column 3.
- LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Find an equation relating i and t. First we try plotting i versus t. But this only produces a curve as shown to the right and from this curve it is difficult to find the equation.
This does produce a straight line as shown to the right and we know how to find the equation of a straight line. To find the equation of this line we will pretend for a moment that the axes are labelled y and x. To determine m and b we will use the elimination method discussed in section 6. We can solve them by elimination.