### GRAPHS OF LOGARITHMIC FUNCTIONS

In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. We will also illustrate how you can use graphs to HELP you solve logarithmic problems.

Solving an equation from a graph. When we solve an equation algebraically, we set the equation equal to zero and find those values that cause the equation to equal zero. When we solve an equation graphically, we look for points of intersection, or re-arrange the equation and look for the x-intercepts (best method).

If there is one equation, we look to see where the graph crosses the x-axis. The x-intercepts are the solutions to the equation. The x-intercepts are those values of x that cause the function value to be zero.

If you are solving two equations graphically, you have two options. You can graph both equations and determine the value of x at the point(s) of intersection. You can also create a new graph by subtracting one function from another, graph the new function, and find the x-intercepts.

In the following examples, we will solve the equations algebraically and graphically.

Example 19: Solve the equation for x algebraically and graphically.

Algebraically:

1. Since you cannot take the logarithm of a negative number, there is a constraint on the values of the x’s. For the term to be valid, x > 6. For the term to be valid, x > -6. IF we restrict the values of x to be greater than zero, both terms are valid.
2. Simplify the equation to read .
3. Convert the equation to an exponential equation with base e.
4. Solve for x: Add 36 to both sides of the equation and we have . Take the square roots of both sides of the equation and we have .
5. Note that only one of these solutions is > 6. Therefore the solution is x = 13.579881.

Graphically - Method 1:

1. Graph and graph on the same coordinate axis and find the point(s), if any, of intersection.
2.

3. The graphs intersect at one point. (13.579881, 5). The solution is the value of x, or x = 13.579881.

Graphically - Method 2:

1. Subtract 5 from both sides of the equation and we have the equation
2. Call the left side of the equation f(x) and graph f(x). Call the right side of the equation h(x) and graph h(x). Since the right side of the equation is nothing more than the x-axis, you will be looking for the x intercepts on the graphs of .
3.

4. The graph of f(x) crosses the x-axis at x = 13.579881.
5. The solution is x = 13.579881.

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