Exponential Equations with Unlike Bases

Learning Outcomes

  • Use logarithms to solve exponential equations whose terms cannot be rewritten with the same base
  • Solve exponential equations of the form  [latex]y=A{e}^{kt}[/latex] for t
  • Recognize when there may be extraneous solutions or no solutions for exponential equations

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since [latex]\mathrm{log}\left(a\right)=\mathrm{log}\left(b\right)[/latex] can be rewritten as a = b, we may apply logarithms with the same base on both sides of an exponential equation.

In our first example we will use the laws of logs combined with factoring to solve an exponential equation whose terms do not have the same base. Note how we rewrite the exponential terms as logarithms first.

Example

Solve [latex]{5}^{x+2}={4}^{x}[/latex].

In general we can solve exponential equations whose terms do not have like bases in the following way:

  1. Apply the logarithm to both sides of the equation.
    • If one of the terms in the equation has base [latex]10[/latex], use the common logarithm.
    • If none of the terms in the equation has base [latex]10[/latex], use the natural logarithm.
  2. Use the rules of logarithms to solve for the unknown.

The following video provides more examples of solving exponential equations.

Think About It

Is there any way to solve [latex]{2}^{x}={3}^{x}[/latex]?

Use the text box below to formulate an answer or example before you look at the solution.

 Equations Containing [latex]e[/latex]

Base e is a very common base found in science, finance, and engineering applications. When we have an equation with a base e on either side, we can use the natural logarithm to solve it. Earlier, we introduced a formula that models continuous growth, [latex]y=A{e}^{kt}[/latex]. This formula is found in business, finance, and many biological and physical science applications. In our next example, we will show how to solve this equation for [latex]t[/latex], the elapsed time for the behavior in question.

Example

Solve [latex]100=20{e}^{2t}[/latex].

Analysis of the Solution

Using laws of logs, we can also write this answer in the form [latex]t=\mathrm{ln}\sqrt{5}[/latex]. If we want a decimal approximation of the answer, we use a calculator.

Think About It

Does every equation of the form [latex]y=A{e}^{kt}[/latex] have a solution? Write your thoughts or an example in the textbox below before you check the answer.

We will provide one more example using the continuous growth formula, but this time we have to do a couple steps of algebra to get it in a form that can be solved.

Example

Solve [latex]4{e}^{2x}+5=12[/latex].

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 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 equation.

Example

Solve [latex]{e}^{2x}-{e}^{x}=56[/latex].

 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 [latex]{e}^{x}=-7[/latex] because a positive number never equals a negative number. The solution [latex]x=\mathrm{ln}\left(-7\right)[/latex] 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.

Summary

The inverse operation of exponentiation is the logarithm, so we can use logarithms to solve exponential equations whose terms do not have the same bases. This is similar to using multiplication to “undo” division or addition to “undo” subtraction. It is important to check exponential equations for extraneous solutions or no solutions.