4.6: Exponential and Logarithmic Equations
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Learning Objectives
- Solve exponential equations by rewriting with a common base, or rewriting in logarithmic form.
- Solve logarithmic equations by rewriting in exponential form or using the one-to-one property of logarithms.
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.
Solve Exponential Equations
Use the One-to-One Property of Exponential Functions
The first technique 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 b>0, b≠1, bS=bT 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 34x−7=32x3. 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:
34x−7=32x334x−7=32x31Rewrite 3 as 3134x−7=32x−1Use the division property of exponents4x−7=2x−1Apply the one-to-one property of exponents2x=6Subtract 2x and add 7 to both sidesx=3Divide by 3
THE 1-1 PROPERTY OF EXPONENTIAL FUNCTIONS
For any algebraic expressions S and T, and any positive real number b≠1,
bS=bT if and only if S=T
How to: Solve an exponential equation of the form bS=bT, where S and T are algebraic expressions.
- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form bS=bT.
- Use the one-to-one property to set the exponents equal.
- Solve the resulting equation, S=T, for the unknown.
Example 4.6.1: Solve an Exponential Equation with a Common Base
Solve 2x−1=22x−4.
Solution
2x−1=22x−4The common base is 2x−1=2x−4By the one-to-one property the exponents must be equalx=3Solve for x
Try It 4.6.1
Solve 52x=53x+2.
- Answer
-
x=−2
Common Base Method
Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation 256=4x−5. We can rewrite both sides of this equation as a power of 2. Then we apply the rules of exponents, along with the one-to-one property, to solve for x:
256=4x−528=(22)x−5Rewrite each side as a power with base 228=22x−10Use the one-to-one property of exponents8=2x−10Apply the one-to-one property of exponents18=2xAdd 10 to both sidesx=9Divide by 2
How to: Given an exponential equation with unlike bases, use the one-to-one property to solve it.
- Rewrite each side in the equation as a power with a common base.
- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form bS=bT.
- Use the one-to-one property to set the exponents equal.
- Solve the resulting equation, S=T, for the unknown.
Example 4.6.2: Solve Equations by Rewriting Them to Have a Common Base
Solve 8x+2=16x+1.
Solution
8x+2=16x+1(23)x+2=(24)x+1Write 8 and 16 as powers of 223x+6=24x+4To take a power of a power, multiply exponents3x+6=4x+4Use the one-to-one property to set the exponents equalx=2Solve for x
Try It 4.6.2
Solve 52x=253x+2.
- Answer
-
x=−1
4.6.3
Solve 25x=√2.
Solution
25x=212Write the square root of 2 as a power of 25x=12Use the one-to-one propertyx=110Solve for x
Try It 4.6.3
Solve 5x=√5.
- Answer
-
x=12
Do all exponential equations have a solution? How can we tell if there is not a solution?
No. Recall that the range of an exponential function is always positive. In the process of solving an exponential equation, if the equation obtained is an exponential expression that is not equal to a positive number, there is no solution for that equation.
Example 4.6.4: Exponential Equation with no solution
Solve 3x+1=−2.
Solution
This equation has no solution. There is no real value of x that will make the equation a true statement because any power of a positive number is positive. The figure below shows that the two graphs do not cross so the left side of the equation is never equal to the right side. Thus the equation has no solution.

Try It 4.6.4
Solve 2x=−100.
- Answer
-
The equation has no solution.
Rewrite in Logarithmic Form
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 log(a)=log(b) is equivalent to a=b, we may apply logarithms with the same base on both sides of an exponential equation.
How to: Solve an exponential equation in which a common base cannot be found
- Apply the logarithm to both sides of the equation.
- If one of the terms in the equation has base 10, use the common logarithm.
- If one of the terms in the equation has base e, use the natural logarithm.
- Use the rules of logarithms to solve for the unknown.
Example 4.6.5: Solve an Equation Containing Powers of Different Bases
Solve 5x+2=4x.
Solution
5x+2=4xThere is no easy way to get the powers to have the same baseln5x+2=ln4xTake ln of both sides(x+2)ln5=xln4Power Rule for Logarithmsxln5+2ln5=xln4Use the distributive lawxln5−xln4=−2ln5Get terms containing x on one side, terms without x on the otherx(ln5−ln4)=−2ln5On the left hand side, factor out an xxln(54)=ln(125)Power and Quotient Rules for Logarithmsx=ln(125)ln(54)Divide by the coefficient of x
Try It 4.6.5
Solve 2x=3x+1.
- Answer
-
x=ln3ln(23)
Is there any way to solve 2x=3x?
Yes. The solution is x=0. (Take the ln of both sides, use the power rule, and solve for x).
Equations Containing e
One common type of exponential equations are those with base e. This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base e on either side, we can use the natural logarithm to solve it.
How to: Given an equation of the form y=Aekt, solve for t.
- Divide both sides of the equation by A.
- Apply the natural logarithm of both sides of the equation.
- Divide both sides of the equation by k.
Example 4.6.6: Solve an Equation of the Form y=Aekt
Solve 100=20e2t.
Solution
100=20e2t5=e2tDivide by the coefficient of the powerln5=2tTake ln of both sides. Use the fact that ln(x) and ex are inverse functionst=ln52Divide by the coefficient of t
Analysis
Using logarithm rules, this answer can be rewritten in the form t=ln√5. A calculator can be used to obtain a decimal approximation of the answer, t≈0.8047.
Try It 4.6.6
Solve 3e0.5t=11.
- Answer
-
t=2ln(113) or ln(113)2
Does every equation of the form y=Aekt have a solution?
No. When k≠0, there is a solution when y and A are either both 0, or when neither is 0 and they have the same sign. An example of an equation with this form that does not have a solution is 2=−3et, which would mean et is negative, which is impossible.
Example 4.6.7: Solve an Equation That Can Be Simplified to the Form y=Aekt
Solve 4e2x+5=12.
Solution
4e2x+5=124e2x=7Combine like termse2x=74Divide by the coefficient of the power2x=ln(74)Take ln of both sides and use lneu=ux=12ln(74)Solve for x
Try It 4.6.7
Solve 3+e2t=7e2t.
- Answer
-
t=ln(1√2)=−12ln(2)
Use properties of exponents
Being able to solve equations of the form y=Aekt suggests a final way of solving exponential equations that can be rewritten in the form a=bp(x). We will redo example 5 using this alternate method. The method used in example 5 is good practice using log properties. This alternative approach uses exponent properties instead.
How to: Solve an exponential equation in which a common base cannot be found
- If possible, use Rules of Exponents to write the equation in the form a=bp(x).
- Rewrite the exponential equation in its logarithmic form: p(x)=logb(a), and solve for x.
Example 4.6.8: Simplify using Exponent Rules before writing in Logarithmic form
Solve 5x+2=4x.
Solution
5x+2=4xUse the Exponent Product Rule in reverse5x⋅52=4x 5x⋅25=4x 25=4x5xLike Powers Rule 25=(45)xRewrite as a log equationx=log45(25)Now use the change of base rulex=ln25ln45≈−14.4251
Exponential Equations that are quadratic in form
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 value of the argument of a logarithm is negative, there is no output.
Example 4.6.9: Solve Exponential Functions that are Quadratic in Form
Solve e2x−ex=56.
Solution:
e2x−ex=56e2x−ex−56=0Get one side of the equation equal to zero(ex+7)(ex−8)=0Factor by the FOIL methodex+7=0ex=−7x=undefinedex−8=0ex=8x=ln8Use the Zero Factor Property to solve for xRewrite in log form Reject the equation in which the power equals a negative number
Analysis
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 ex=−7 because a positive number never equals a negative number. The solution ln(−7) is not a real number, and in the real number system this solution is rejected as an extraneous solution.
Try It 4.6.9
Solve e2x=ex+2.
- Answer
-
x=ln2
Solving Logarithmic Equations
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.
Rewrite in Exponential Form
We have already seen that every logarithmic equation logb(x)=y is equivalent to the exponential equation by=x. 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 log2(2)+log2(3x−5)=3. To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then rewrite the logarithmic equation in exponential form to solve for x:
log2(2)+log2(3x−5)=3log2(2(3x−5))=3Apply the product rule of logarithmslog2(6x−10)=3Distribute23=6x−10Apply the definition of a logarithm8=6x−10Calculate 2318=6xAdd 10 to both sidesx=3Divide by 6, then check the solution!
Whenever solving an equation with logarithms, it is always necessary to check that the solution is in the domain of the original equation. If it is not, it must be rejected as a solution.
USE THE DEFINITION OF A LOGARITHM TO SOLVE LOGARITHMIC EQUATIONS
For any algebraic expression S and real numbers b and c, where b>0, b≠1,
logb(S)=c if and only if bc=S
Example 4.6.10: Rewrite a Logarithmic Equation in Exponential Form
Solve 2lnx+3=7.
Solution
2lnx+3=72lnx=4Subtract 3lnx=2Divide by 2x=e2Rewrite in exponential form
Try It 4.6.10
Solve 6+lnx=10.
- Answer
-
x=e4
Example 4.6.11
Solve 2ln(6x)=7.
Solution:
2ln(6x)=7ln(6x)=72Divide by 26x=e(72)Use the definition of lnx=16e(72)Divide by 6
Try It 4.6.11
Solve 2ln(x+1)=10.
- Answer
-
x=e5−1
Use the One-to-One Property of Logarithms
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 S>0, T>0 and any positive real number b, where b≠1,
logbS=logbT if and only if S=T.
For example,
If log2(x−1)=log2(8), then x−1=8.
So, if x−1=8, then we can solve for x, and we get x=9. To check, we can substitute x=9 into the original equation: log2(9−1)=log2(8)=3. 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 log(3x−2)−log(2)=log(x+4). 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:
log(3x−2)−log(2)=log(x+4)log(3x−22)=log(x+4)Apply the quotient rule of logarithms3x−22=x+4Apply the one to one property of a logarithm3x−2=2x+8Multiply both sides of the equation by 2x=10Subtract 2x and add 2
To check the result, substitute x=10 into log(3x−2)−log(2)=log(x+4).
log(3(10)−2)−log(2)=log((10)+4)log(28)−log(2)=log(14)log(282)=log(14)The solution checks
USE 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 b≠1,
logbS=logbT if and only if S=T
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 logbS=logbT.
- Use the one-to-one property to set the arguments equal.
- Solve the resulting equation, S=T, for the unknown.
- Confirm that each solution is correct. If a solution when substituted in the original equation makes one of the log arguments zero or negative, that solution must be rejected.
Example 4.6.12: Use the One-to-One Property of Logarithms to Solve an Equation
Solve ln(x2)=ln(2x+3).
Solution
\begin{align*} \ln(x^2)&= \ln(2x+3)\\ x^2&= 2x+3 \qquad &&\text{Use the one-to-one property of the logarithm}\\ x^2-2x-3&= 0 \qquad &&\text{Get zero on one side before factoring}\\ (x-3)(x+1)&= 0 \qquad &&\text{Factor using FOIL}\\ x-3 = 0 \qquad \quad & \text{or} \qquad x+1=0 &&\text{If a product is zero, one of the factors must be zero}\\ x=3 \qquad \quad &\quad \qquad x= -1 \qquad &&\text{Solve for x}\\ 3^2 >0 \text{ and } 2(3)+3 > 0 \color{Cerulean}{✓} \quad & \quad (-1)^2 > 0 \text{ and } 2(-1)+3 > 0 \color{Cerulean}{✓} && \text{Check the solution when substituted in the arguments is }> 0 \end{align*}
Analysis
There are two solutions: 3 or −1. The solution −1 is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.
Try It \PageIndex{12}
Solve \ln(x^2)=\ln1.
- Answer
-
x=1 or x=−1
Key Equations
One-to-one property for exponential functions | For any algebraic expressions S and T and any positive real number b, where b^S=b^T if and only if S=T. |
Definition of a logarithm | For any algebraic expression S and positive real numbers b and c, where b≠1, {\log}_b(S)=c if and only if b^c=S. |
One-to-one property for logarithmic functions | For any algebraic expressions S and T and any positive real number b, where b≠1, {\log}_bS={\log}_bT 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 and then using the fact that \ln (e^U) = U .
- After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions (Specifically, b^p is ALWAYS positive).
- When given an equation of the form {\log}_b(S)=c, where S is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation b^c=S, and solve for the unknown.
- When given an equation of the form {\log}_bS={\log}_bT, 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. Check that the answers do not make any original log arguments zero or negative.
Contributors
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.