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Solving Logarithmic Equations
Deciding How to Solve Logarithmic Equation
When asked to solve a logarithmic equation such as or the first thing we need to decide is how to solve the problem. Some logarithmic problems are solved by simply dropping the logarithms while others are solved by rewriting the logarithmic problem in exponential form. How do we decide what is the correct way to solve a logarithmic problem? The key is to look at the problem and decide if the problem contains only logarithms or if the problem has terms without logarithms.
If we consider the problem this problem contains a term, 5, that does not have a logarithm. So, the correct way to solve these types of logarithmic problems is to rewrite the logarithmic problem in exponential form. If we consider the example this problem contains only logarithms. So, the correct way to solve these types of logarithmic problems is to simply drop the logarithms.
Properties of Logarithms Revisited
When solving logarithmic equation, we may need to use the properties of logarithms to simplify the problem first. The properties of logarithms are listed below as a reminder.
Solving Logarithmic Equations Containing Only Logarithms
After observing that the logarithmic equation contains only logarithms, what is the next step?
This statement says that if an equation contains only two logarithms, on opposite sides of the equal sign, with the same base then the problem can be solved by simply dropping the logarithms.
Properties for Condensing Logarithms Property 1: 0 = log 1a – Zero Exponent Rule
Property 2: 1 =log aa
Property 3: loga x + loga y = log (xy)a – Product Rule
Property 4: (^) a a a
x log x log y log y
Ê ˆ
- = Á ˜
Ë ¯
Property 5: y log (^) a x = log (^) axy– Power Rule
OneT o One Property of Logarithms
If log (^) b M = log (^) bN,then M = N.
log (5x 2 + 7) = 5 log (7x 3 + 3) = log (5x 3 +9),
log (5x 2 + 7) =5,
log (7x 3 + 3) = log (5x 3 +9),
Let’s finish solving the problem log (7x 3 + 3) = log (5x 3 +9).
Since the problem has only two logarithms on opposite sides of the equal sign, the problem can be solved by dropping the logarithms.
Drop the logarithms.
Finish solving by subtracting 5x from each side, subtracting 3 from each side, and finally dividing each side by 2.
Check the answer; this is an acceptable answer because we get a positive number when it is plugged back in.
Therefore, the solution to the problem log (7x 3 + 3) = log (5x 3 + 9)is x = 3.
Here is another example, solve log (x 7 - 2) + log (x 7 + 3) =log 14. 7
This problem can be simplified by using Property 3 which changes the addition of logarithms to multiplication.
Drop the logarithms.
Simplify the problem by distributing or FOILing and combining like terms.
Solve the problem by subtracting 14 from each to get it equal to zero, and then factoring or using the quadratic formula to find the values of x.
Check the answers, only one answer is acceptable because the other answer produces a negative number when it is plugged back in.
Therefore, the solution to the problem log (x 7 - 2) + log (x 7 + 3) = log 14 7 is x = 5.
Now that we have looked at a couple of examples of solving logarithmic equations containing only logarithms, let’s list the steps for solving logarithmic equations containing only logarithms.
log (7x 3 + 3) = log (5x 3 +9).
7x + 3 = 5x + 9
x = 3
x = 3
log ((x 7 - 2)(x + 3)) =log 14 7
(x - 2)(x + 3) = 14
x 2 - x - 6 = 14
x 2 x 20 0
(x 4)(x 5) 0
x 4 or x 5
x = 5
Here is another example, solve ln(4x - 1) =3.
ln(4x - 1) = 3 This problem does not need to be simplified because there is only one logarithm in the problem.
4x - 1 = e^3 Rewrite the problem in exponential form by moving the base of the logarithm to the other side. For natural logarithms the base is e.
4x - 1 ª 20.085537 Simplify the problem by cubing e. Round the answer as appropriate, these answers will use 6 decimal places.
x ª 5.271384 Solve for x by adding 1 to each side and then dividing each side by 4.
x ª 5.271384 Check the answer; this is an acceptable answer because we get a positive number when it is plugged back in.
Therefore, the solution to the problem ln(4x - 1) = 3 is x ≈ 5.271384.
Now that we have looked at a couple of examples of solving logarithmic equations containing terms without logarithms, let’s list the steps for solving logarithmic equations containing terms without logarithms.
Steps for Solving Logarithmic Equations Containing Terms without Logarithms
Step 1 : Determine if the problem contains only logarithms. If so, stop and use Steps for Solving Logarithmic Equations Containing Only Logarithms. If not, go to Step 2.
Step 2 : Use the properties of logarithms to simplify the problem if needed. If the problem has more than one logarithm on either side of the equal sign then the problem can be simplified.
Step 3 : Rewrite the problem in exponential form.
Step 4 : Simplify the problem.
Step 5 : Solve for x.
Step 6 : Check your answer(s). Remember we cannot take the logarithm of a negative number, so we need to make sure that when we plug our answer(s) back into the original equation we get a positive number. Otherwise, we must drop that answer(s).
Examples – Now let’s use the steps shown above to work through some examples. These examples will be a mixture of logarithmic equations containing only logarithms and logarithmic equations containing terms without logarithms.
Example 1 : Solve log (9x 3 + 2) = 4
This problem contains terms without logarithms.
This problem does not need to be simplified because there is only one logarithm in the problem.
Rewrite the problem in exponential form by moving the base of the logarithm to the other side.
Simplify the problem by raising 3 to the fourth power.
Solve for x by subtracting 2 from each side and then dividing each side by 9.
Check the answer; this is an acceptable answer because we get a positive number when it is plugged back in.
Therefore, the solution to the problem log (9x 3 + 2) = 4 is
x. 9
Example 2 : Solve log 4 x + log (x 4 - 12) = 3
This problem contains terms without logarithms.
This problem can be simplified by using Property 3 which changes the addition of logarithms to multiplication.
Rewrite the problem in exponential form by moving the base of the logarithm to the other side.
Simplify the problem by distributing and cubing the 4.
Solve the problem by subtracting 64 from each to get it equal to zero, and then factoring or using the quadratic formula to find the values of x.
x = 16 Check the answers, only one answer is acceptable because the other answer produces a negative number when it is plugged back in.
Therefore, the solution to the problem log 4 x + log (x 4 - 12) = 3 is x = 16.
log (9x 3 + 2) = 4
9x + 2 = 34
9x + 2 = 81
79 x 9
x 9
log (x(x 4 - 12)) = 3
x(x - 12) = 43
x 2 - 12x = 64
x 2 12x 64 0
(x 4)(x 16) 0
x 4 or x 16
log 4 x + log (x 4 - 12) = 3
log (9x 3 + 2) = 4
Example 5 : Solve log (x 2 + 1) - log (x 2 - 4) = 3
This problem contains terms without logarithms.
This problem can be simplified by using Property 4 which changes the subtraction of logarithms to division.
Rewrite the problem in exponential form by moving the base of the logarithm to the other side.
Simplify the problem by cubing the 2.
Solve for x by cross multiplying, distributing, subtracting 8x from each side, subtracting 1 from each side, and finally dividing each side by –7.
Check the answer; this is an acceptable answer because we get a positive number when it is plugged back in.
Therefore, the solution to the problem log (x 2 + 1) - log (x 2 - 4) = 3 is
x. 7
Example 6 : Solve log (x 6 + 4) + log (x 6 - 2) =log (4x) 6
This problem contains only logarithms.
This problem can be simplified by using Property 3 which changes the addition of logarithms to multiplication.
Drop the logarithms.
Simplify the problem by distributing or FOILing and combining like terms.
Solve the problem by subtracting 4x from each to get it equal to zero, and then factoring or using the quadratic formula to find the values of x.
Check the answers, only one answer is acceptable because the other answer produces a negative number when it is plugged back in.
Therefore, the solution to the problem log (x 6 + 4) + log (x 6 - 2) = log (4x) 6 is x = 4.
2
x 1 log 3 x 4
Ê + ˆ
Á ˜=
Ë - ¯
x (^1 ) x 4
x 1 8 x 4
x 1 8(x 4)
x 1 8x 32
33 x 7
x 7
log (x 2 + 1) - log (x 2 - 4) = 3
log (x 6 + 4) + log (x 6 - 2) =log (4x) 6
log ((x 6 + 4)(x - 2)) =log (4x) 6
(x + 4)(x - 2) =4x
x 2 + 2x - 8 =4x
x 2 2x 8 0
(x 2)(x 4) 0
x 2 or x 4
x = 4
Addition Examples
If you would like to see more examples of solving logarithmic equations, just click on the link below.
Additional Examples
Practice Problems
Now it is your turn to try a few practice problems on your own. Work on each of the problems below and then click on the link at the end to check your answers.
Problem 1 : Solve: log (4x 5 + 11) = 2
Problem 2 : Solve: log (x 2 + 5) - log (2x 2 - 1) = 5
Problem 3 : Solve: log x 8 + log (x 8 + 6) = log (5x 8 +12)
Problem 4 : Solve: log 6 x + log (x 6 - 9) = 2
Problem 5 : Solve: ln(6x - 5) = 3
Problem 6 : Solve: log (3x 4 - 2) - log (4x 4 + 1) = 2
Problem 7 : Solve: log (x 3 2 - 6x) = 3
Problem 8 : Solve: log(x - 2) - log(2x - 3) =log 2
Solutions to Practice Problems
