Exponential Equations
Learning Objectives
- Solve an exponential equation with a common base
- Rewrite an exponential equation so all terms have a common base, then solve
- Recognize when an exponential equation does not have a solution
- Use logarithms to solve exponential equations
[latex]\begin{array}{l}{3}^{4x - 7}\hfill & =\frac{{3}^{2x}}{3}\hfill & \hfill \\ {3}^{4x - 7}\hfill & =\frac{{3}^{2x}}{{3}^{1}}\hfill & {\text{Rewrite 3 as 3}}^{1}.\hfill \\ {3}^{4x - 7}\hfill & ={3}^{2x - 1}\hfill & \text{Use the division property of exponents}\text{.}\hfill \\ 4x - 7\hfill & =2x - 1\text{ }\hfill & \text{Apply the one-to-one property of exponents}\text{.}\hfill \\ 2x\hfill & =6\hfill & \text{Subtract 2}x\text{ and add 7 to both sides}\text{.}\hfill \\ x\hfill & =3\hfill & \text{Divide by 3}\text{.}\hfill \end{array}[/latex]
A General Note: Using the One-to-One Property of Exponential Functions to Solve Exponential Equations
For any algebraic expressions S and T, and any positive real number [latex]b\ne 1[/latex], [latex-display]{b}^{S}={b}^{T}\text{ if and only if }S=T[/latex-display]How To: Given an exponential equation with the form [latex]{b}^{S}={b}^{T}[/latex], where S and T are algebraic expressions with an unknown, solve for the unknown.
- 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.
- Solve the resulting equation, S = T, for the unknown.
Example: Solving an Exponential Equation with a Common Base
Solve [latex]{2}^{x - 1}={2}^{2x - 4}[/latex].Answer: [latex-display]\begin{array}{l} {2}^{x - 1}={2}^{2x - 4}\hfill & \text{The common base is }2.\hfill \\ \text{ }x - 1=2x - 4\hfill & \text{By the one-to-one property the exponents must be equal}.\hfill \\ \text{ }x=3\hfill & \text{Solve for }x.\hfill \end{array}[/latex-display]
Try It
Solve [latex]{5}^{2x}={5}^{3x+2}[/latex].Answer: [latex]x=–2[/latex]
Example: Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base
Solve [latex]{2}^{5x}=\sqrt{2}[/latex].Answer: [latex-display]\begin{array}{l}{2}^{5x}={2}^{\frac{1}{2}}\hfill & \text{Write the square root of 2 as a power of }2.\hfill \\ 5x=\frac{1}{2}\hfill & \text{Use the one-to-one property}.\hfill \\ x=\frac{1}{10}\hfill & \text{Solve for }x.\hfill \end{array}[/latex-display]
Try It
Solve [latex]{5}^{x}=\sqrt{5}[/latex].Answer: [latex]x=\frac{1}{2}[/latex]
Use logarithms to solve 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] is equivalent to a = b, we may apply logarithms with the same base on both sides of an exponential equation.How To: Given an exponential equation in which a common base cannot be found, solve for the unknown.
- Apply the logarithm of both sides of the equation.
- If one of the terms in the equation has base 10, use the common logarithm.
- If none of the terms in the equation has base 10, use the natural logarithm.
- Use the rules of logarithms to solve for the unknown.
Example: Solving an Equation Containing Powers of Different Bases
Solve [latex]{5}^{x+2}={4}^{x}[/latex].Answer: [latex-display]\begin{array}{l}\text{ }{5}^{x+2}={4}^{x}\hfill & \text{There is no easy way to get the powers to have the same base}.\hfill \\ \text{ }\mathrm{ln}{5}^{x+2}=\mathrm{ln}{4}^{x}\hfill & \text{Take ln of both sides}.\hfill \\ \text{ }\left(x+2\right)\mathrm{ln}5=x\mathrm{ln}4\hfill & \text{Use laws of logs}.\hfill \\ \text{ }x\mathrm{ln}5+2\mathrm{ln}5=x\mathrm{ln}4\hfill & \text{Use the distributive law}.\hfill \\ \text{ }x\mathrm{ln}5-x\mathrm{ln}4=-2\mathrm{ln}5\hfill & \text{Get terms containing }x\text{ on one side, terms without }x\text{ on the other}.\hfill \\ x\left(\mathrm{ln}5-\mathrm{ln}4\right)=-2\mathrm{ln}5\hfill & \text{On the left hand side, factor out an }x.\hfill \\ \text{ }x\mathrm{ln}\left(\frac{5}{4}\right)=\mathrm{ln}\left(\frac{1}{25}\right)\hfill & \text{Use the laws of logs}.\hfill \\ \text{ }x=\frac{\mathrm{ln}\left(\frac{1}{25}\right)}{\mathrm{ln}\left(\frac{5}{4}\right)}\hfill & \text{Divide by the coefficient of }x.\hfill \end{array}[/latex-display]
Try It
Solve [latex]{2}^{x}={3}^{x+1}[/latex].Answer: [latex]x=\frac{\mathrm{ln}3}{\mathrm{ln}}\left(23\right)[/latex]
Q & A
Does every equation of the form [latex]y=A{e}^{kt}[/latex] have a solution?
No. There is a solution when [latex]k\ne 0[/latex], 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 [latex]2=-3{e}^{t}[/latex].Example: Solving an Equation That Can Be Simplified to the Form [latex]y=A{e}^{kt}[/latex]
Solve [latex]4{e}^{2x}+5=12[/latex].Answer: [latex-display]\begin{array}{l}4{e}^{2x}+5=12\hfill & \hfill \\ 4{e}^{2x}=7\hfill & \text{Combine like terms}.\hfill \\ {e}^{2x}=\frac{7}{4}\hfill & \text{Divide by the coefficient of the power}.\hfill \\ 2x=\mathrm{ln}\left(\frac{7}{4}\right)\hfill & \text{Take ln of both sides}.\hfill \\ x=\frac{1}{2}\mathrm{ln}\left(\frac{7}{4}\right)\hfill & \text{Solve for }x.\hfill \end{array}[/latex-display]
Try It
Solve [latex]3+{e}^{2t}=7{e}^{2t}[/latex].Answer: [latex]t=\mathrm{ln}\left(\frac{1}{\sqrt{2}}\right)=-\frac{1}{2}\mathrm{ln}\left(2\right)[/latex]
Q & A
Does every logarithmic equation have a solution?
No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.Licenses & Attributions
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