Exponential and Logarithmic Equations
In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions. Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.Exponential Equations
The first technique we will introduce for solving exponential equations involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, S, and T, where [latex]b>0,\text{ }b\ne 1[/latex], [latex]{b}^{S}={b}^{T}[/latex] if and only if S = T. In other words, 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. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown. For example, consider the equation [latex]{3}^{4x - 7}=\frac{{3}^{2x}}{3}[/latex]. To solve for x, we use the division property of exponents to rewrite the right side so that both sides have the common base, 3. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x:[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.Logarithmic Equations
We have already seen that every logarithmic equation [latex]{\mathrm{log}}_{b}\left(x\right)=y[/latex] is equivalent to the exponential equation [latex]{b}^{y}=x[/latex]. We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression. For example, consider the equation [latex]{\mathrm{log}}_{2}\left(2\right)+{\mathrm{log}}_{2}\left(3x - 5\right)=3[/latex]. To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for x:[latex]\begin{array}{l}{\mathrm{log}}_{2}\left(2\right)+{\mathrm{log}}_{2}\left(3x - 5\right)=3\hfill & \hfill \\ \text{ }{\mathrm{log}}_{2}\left(2\left(3x - 5\right)\right)=3\hfill & \text{Apply the product rule of logarithms}.\hfill \\ \text{ }{\mathrm{log}}_{2}\left(6x - 10\right)=3\hfill & \text{Distribute}.\hfill \\ \text{ }{2}^{3}=6x - 10\hfill & \text{Apply the definition of a logarithm}.\hfill \\ \text{ }8=6x - 10\hfill & \text{Calculate }{2}^{3}.\hfill \\ \text{ }18=6x\hfill & \text{Add 10 to both sides}.\hfill \\ \text{ }x=3\hfill & \text{Divide by 6}.\hfill \end{array}[/latex]
A General Note: Using the Definition of a Logarithm to Solve Logarithmic Equations
For any algebraic expression S and real numbers b and c, where [latex]b>0,\text{ }b\ne 1[/latex],[latex]{\mathrm{log}}_{b}\left(S\right)=c\text{if and only if}{b}^{c}=S[/latex]
Example: Using Algebra to Solve a Logarithmic Equation
Solve [latex]2\mathrm{ln}x+3=7[/latex].Answer: [latex-display]\begin{array}{l}2\mathrm{ln}x+3=7\hfill & \hfill \\ \text{ }2\mathrm{ln}x=4\hfill & \text{Subtract 3}.\hfill \\ \text{ }\mathrm{ln}x=2\hfill & \text{Divide by 2}.\hfill \\ \text{ }x={e}^{2}\hfill & \text{Rewrite in exponential form}.\hfill \end{array}[/latex-display]
Try It
Solve [latex]6+\mathrm{ln}x=10[/latex].Answer: [latex]x={e}^{4}[/latex]
Example: Using a Graph to Understand the Solution to a Logarithmic Equation
Solve [latex]\mathrm{ln}x=3[/latex].Answer: [latex-display]\begin{array}{l}\mathrm{ln}x=3\hfill & \hfill \\ x={e}^{3}\hfill & \text{Use the definition of the natural logarithm}\text{.}\hfill \end{array}[/latex-display] Below is a graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words [latex]{e}^{3}\approx 20[/latex]. A calculator gives a better approximation: [latex]{e}^{3}\approx 20.0855[/latex].
The graphs of [latex]y=\mathrm{ln}x[/latex] and y = 3 cross at the point [latex]\left(e^3,3\right)[/latex], which is approximately (20.0855, 3).Try It
Use a graphing calculator to estimate the approximate solution to the logarithmic equation [latex]{2}^{x}=1000[/latex] to 2 decimal places.Answer: [latex]x\approx 9.97[/latex]
Use the one-to-one property of logarithms to solve logarithmic equations
As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers x > 0, S > 0, T > 0 and any positive real number b, where [latex]b\ne 1[/latex],[latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T\text{ if and only if }S=T[/latex].
For example,[latex]\text{If }{\mathrm{log}}_{2}\left(x - 1\right)={\mathrm{log}}_{2}\left(8\right),\text{then }x - 1=8[/latex].
So, if [latex]x - 1=8[/latex], then we can solve for x, and we get x = 9. To check, we can substitute x = 9 into the original equation: [latex]{\mathrm{log}}_{2}\left(9 - 1\right)={\mathrm{log}}_{2}\left(8\right)=3[/latex]. In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown. For example, consider the equation [latex]\mathrm{log}\left(3x - 2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right)[/latex]. To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for x:[latex]\begin{array}{l}\mathrm{log}\left(3x - 2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right)\hfill & \hfill \\ \text{ }\mathrm{log}\left(\frac{3x - 2}{2}\right)=\mathrm{log}\left(x+4\right)\hfill & \text{Apply the quotient rule of logarithms}.\hfill \\ \text{ }\frac{3x - 2}{2}=x+4\hfill & \text{Apply the one to one property of a logarithm}.\hfill \\ \text{ }3x - 2=2x+8\hfill & \text{Multiply both sides of the equation by }2.\hfill \\ \text{ }x=10\hfill & \text{Subtract 2}x\text{ and add 2}.\hfill \end{array}[/latex]
To check the result, substitute x = 10 into [latex]\mathrm{log}\left(3x - 2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(x+4\right)[/latex].[latex]\begin{array}{l}\mathrm{log}\left(3\left(10\right)-2\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(\left(10\right)+4\right)\hfill & \hfill \\ \text{ }\mathrm{log}\left(28\right)-\mathrm{log}\left(2\right)=\mathrm{log}\left(14\right)\hfill & \hfill \\ \text{ }\mathrm{log}\left(\frac{28}{2}\right)=\mathrm{log}\left(14\right)\hfill & \text{The solution checks}.\hfill \end{array}[/latex]
A General Note: Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
For any algebraic expressions S and T and any positive real number b, where [latex]b\ne 1[/latex],[latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T\text{ if and only if }S=T[/latex]
Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.How To: Given an equation containing logarithms, solve it using the one-to-one property.
- Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form [latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T[/latex].
- Use the one-to-one property to set the arguments equal.
- Solve the resulting equation, S = T, for the unknown.
Example: Solving an Equation Using the One-to-One Property of Logarithms
Solve [latex]\mathrm{ln}\left({x}^{2}\right)=\mathrm{ln}\left(2x+3\right)[/latex].Answer: [latex-display]\begin{array}{l}\text{ }\mathrm{ln}\left({x}^{2}\right)=\mathrm{ln}\left(2x+3\right)\hfill & \hfill \\ \text{ }{x}^{2}=2x+3\hfill & \text{Use the one-to-one property of the logarithm}.\hfill \\ \text{ }{x}^{2}-2x - 3=0\hfill & \text{Get zero on one side before factoring}.\hfill \\ \left(x - 3\right)\left(x+1\right)=0\hfill & \text{Factor using FOIL}.\hfill \\ \text{ }x - 3=0\text{ or }x+1=0\hfill & \text{If a product is zero, one of the factors must be zero}.\hfill \\ \text{ }x=3\text{ or }x=-1\hfill & \text{Solve for }x.\hfill \end{array}[/latex-display]
Analysis of the Solution
There are two solutions: x = 3 or x = –1. The solution x = –1 is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.Try It
Solve [latex]\mathrm{ln}\left({x}^{2}\right)=\mathrm{ln}1[/latex].Answer: [latex]x=1[/latex] or [latex]x=–1[/latex]
Key Equations
| One-to-one property for exponential functions | For any algebraic expressions S and T and any positive real number b, where[latex]{b}^{S}={b}^{T}[/latex] if and only if S = T. |
| Definition of a logarithm | For any algebraic expression S and positive real numbers b and c, where [latex]b\ne 1[/latex],[latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex] if and only if [latex]{b}^{c}=S[/latex]. |
| One-to-one property for logarithmic functions | For any algebraic expressions S and T and any positive real number b, where [latex]b\ne 1[/latex], [latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T[/latex] if and only if S = T. |
Key Concepts
- We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
- When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown.
- When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown.
- When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side.
- We can solve exponential equations with base e, by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other.
- After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions.
- When given an equation of the form [latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex], where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation [latex]{b}^{c}=S[/latex], and solve for the unknown.
- We can also use graphing to solve equations with the form [latex]{\mathrm{log}}_{b}\left(S\right)=c[/latex]. We graph both equations [latex]y={\mathrm{log}}_{b}\left(S\right)[/latex] and y = c on the same coordinate plane and identify the solution as the x-value of the intersecting point.
- When given an equation of the form [latex]{\mathrm{log}}_{b}S={\mathrm{log}}_{b}T[/latex], where S and T are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation S = T for the unknown.
- Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm.
Glossary
extraneous solution a solution introduced while solving an equation that does not satisfy the conditions of the original equationLicenses & Attributions
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