Exponential Equations

Learning Outcomes

  • Apply the one-to-one property of exponents to solve an exponential equation
  • Solve exponential equations of the form [latex]y=Ae^{kt} \ \mathrm{for} \ t[/latex]

When an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. We can set the exponents equal to one another and solve for the unknown.

For example, to solve the exponential equation [latex]2^{x} = 8[/latex], we might note that [latex]8[/latex] can be written as [latex]2^{3}[/latex].

[latex]2^x=8[/latex]

[latex]2^x=2^3[/latex]

[latex]x=3[/latex]

In general, we can summarize solving exponential equations whose terms all have the same base in this way:

For any algebraic expressions S and T, and any positive real number [latex]b\ne 1[/latex]

[latex]{b}^{S}={b}^{T}\text{ if and only if }S=T[/latex]
  • Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form [latex]{b}^{S}={b}^{T}[/latex].
  • Use the one-to-one property to set the exponents equal to each other.
  • Solve the resulting equation, = T, for the unknown.

Example

Solve [latex]3^{x-1}=81[/latex]

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Exponential Equations with Unlike Bases

Sometimes it is not possible to write both sides of an exponential equation as powers of the same base. Base [latex]e[/latex] is a very common base found in science, finance, and engineering applications. When we have an equation with a base [latex]e[/latex] on either side, we can use the natural logarithm to solve it. Earlier, we introduced a formula that models continuous growth/decay, [latex]y=Ae^{kt}[/latex]. To solve this equation for [latex]t[/latex],

  • Isolate the exponential expression.
  • Use the natural logarithm function to write the exponential equation in logarithmic form.
  • Solve for [latex]t[/latex].

Example

Solve [latex]300=5e^{2t}[/latex].

Example

Solve [latex]0.6=0.1+e^{\frac{t}{2}}[/latex].

Sometimes an equation may not have a solution. For example, to solve [latex]e^{t}= -2[/latex] we might convert to logarithmic form to obtain [latex]t=\mathrm{ln} \ (-2)[/latex]. Entering this on your calculator produces an error message. We can’t take a logarithm of a negative number. The original expression, [latex]e^{t}= -2[/latex] is never true since [latex]e^{t}>0[/latex] for all real numbers [latex]t[/latex].

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