4.4 - Solving Exponential and Logarithm Equations

Inverses

Remember that exponential and logarithmic functions are one-to-one functions. That means that they have inverses. Also recall that when inverses are composed with each other, they inverse out and only the argument is returned. We're going to use that to our benefit to help solve logarithmic and exponential equations.

Please recall the following facts:

• log_a a^x = x
• log 10^x = x
• ln e^x = x
• a ^{log_a x} = x
• 10^{log x} = x
• e^{ln x} = x

Solving Exponential Equations Algebraically

1. Isolate the exponential expression on one side.
2. Take the logarithm of both sides. The base for the logarithm should be the same as the base in the exponential expression. Alternatively, if you are only interested in a decimal approximation, you may take the natural log or common log of both sides (in effect using the change of base formula)
3. Solve for the variable.
4. Check your answer. It may be possible to get answers which don't check. Usually, the answer will involve complex numbers when this happens, because the domain of an exponential function is all reals.

Solving Logarithmic Equations Algebraically

1. Use properties of logarithms to combine the sum, difference, and/or constant multiples of logarithms into a single logarithm.
2. Apply an exponential function to both sides. The base used in the exponential function should be the same as the base in the logarithmic function. Another way of performing this task is to write the logarithmic equation in exponential form.
3. Solve for the variable.
4. Check your answer. It may be possible to introduce extraneous solutions. Make sure that when you plug your answer back into the arguments of the logarithms in the original equation, that the arguments are all positive. Remember, you can only take the log of a positive number.

Solving Equations Graphically

Sometimes, it is impossible to solve an equation involving logarithms or exponential functions algebraically. This is especially true when the equation involves transcendental (logs and/or exponentials) and algebraic components. In cases like these, it may be necessary to use the graphing calculator to help find the solution to the equation.

1. Rewrite the equation so that all the terms are on one side
2. Graph the expression
3. Use the Root or Zero function under the Calc menu.