Read: Exponential Equations with Unlike Bases
Learning Objectives
- Use logarithms to solve exponential equations whose terms cannot be rewritten with the same base
- Solve exponential equations of the form for t
- Recognize when there may be extraneous solutions, or no solutions for exponential equations
Example
Solve .Answer: There is no easy way to get the powers to have the same base for this equation.
- Apply the logarithm of both sides of the equation.
- If one of the terms in the equation has base , use the common logarithm.
- If none of the terms in the equation has base , use the natural logarithm.
- Use the rules of logarithms to solve for the unknown.
Think About It
Is there any way to solve ? Use the textbox below to formulate an answer or example before you look at the solution. [practice-area rows="1"][/practice-area]Answer: Yes. The solution is .
Equations Containing
Example
Solve .Answer:
Extraneous Solutions
Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.
In the next example we will solve an exponential equation that is quadratic in form. We will factor first and then use the zero product principle. Note how we find two solutions, but reject one that does not satisfy the original equaiton.Example
Solve .Answer:
Analysis of the Solution
When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation because a positive number never equals a negative number. The solution is not a real number, and in the real number system this solution is rejected as an extraneous solution.
Think About It
Does every logarithmic equation have a solution? Write your ideas, or a counter example in the box below. [practice-area rows="1"][/practice-area]Answer: No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.
Analysis of the Solution
Using laws of logs, we can also write this answer in the form t=ln5. If we want a decimal approximation of the answer, we use a calculator.Think About It
Does every equation of the form y=Aekt have a solution? Write your thoughts or an example in the textbox below before you check the answer. [practice-area rows="1"][/practice-area]Answer: No. There is a solution when k=0, and when y and A are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is 2=−3et.
Example
Solve 4e2x+5=12.Answer: 4e2x+5=124e2x=7e2x=472x=ln(47)x=21ln(47)Combine like terms.Divide by the coefficient of the power.Take ln of both sides.Solve for x.