If we are given an equation with a logarithm of the same base on both sides we may simply equate the arguments.
Step 1: Use the rules of exponents to isolate a logarithmic expression (with the same base) on both sides of the equation. Step 2: Set the arguments equal to each other. Step 3: Solve the resulting equation. Step 4: Check your answers.
Be sure to check to see if the solutions that you obtain solve the original logarithmic equation. In this study guide we will put a check mark next to the solution after we determine that it really does solve the equation. This process sometimes results in extraneous solutions so we must check our answers. Solve.
Of course, equations like these are very special. Most of the problems that we will encounter will not have a logarithm on both sides. The steps for solving them follow.
Step 1: Use the properties of the logarithm to isolate the log on one side. Step 2: Apply the definition of the logarithm and rewrite it as an exponential equation. Step 3: Solve the resulting equation. Step 4: Check your answers.
If the answer to the logarithmic equation makes the argument negative then it is extraneous. This does not preclude negative answers. We must be sure to check all of our solutions.
Tip: Not all negative solutions are extraneous! Look at the previous set of problems and see that some have negative answers. The check mark indicates that we actually plugged the answers in to see that they do indeed solve the original. Please do not skip this step, extraneous solutions occur often.
One way to graph logarithmic functions is to first graph its inverse exponential. Then use your knowledge about the symmetry of inverses to graph the logarithm.
Recall that inverses are symmetric about the line y = x. For example,
Step 1: Find some points on the exponential f(x). The more points we plot the better the graph will look.
Step 2: Switch the x and y values to obtain points on the inverse.
Step 3: Determine the asymptote.
In practice, we use a combination of techniques to graph logarithms. We can use our knowledge of transformations, techniques for finding intercepts, and symmetry to find as many points as we can to make these graphs. General guidelines follow:
1. Graph the vertical asymptote. All logarithmic functions of the form
have a vertical asymptote at x = h.
2. Find the x- and y-intercepts if they exist. To find x-intercepts set y = f(x) to zero and to find y-intercepts set x = 0.
3. Plot a few more points and graph it.
Graph the following logarithmic functions. State the domain and range.
In the previous solved problem, make a note of the rigid transformations. If we start with the basic graph y = log(x) then the first translation is a shift to the left 3 units y = log(x+3). Next we see a vertical shift up 2 units y = log(x+3)+2 .
In the above problem there was a reflection about the x-axis as well as a shift to the left 3 units.