In this explainer, we will learn how to solve logarithmic equations involving logarithms with different bases.
Let’s first recall the relationship between logarithmic and exponential forms.
Definition: Relationship between Logarithmic and Exponential Forms
For and base , , the logarithmic form is equivalent to the exponential form , which allows us to convert from one form to another once we identify , , and .
An exponential function is the inverse of the logarithmic function . This means if you raise to the power of log of with base or if you raise to the power of first then take the log with base of the result, you get back :
The relationship between logarithmic and exponential forms allows us to deduce the properties satisfied by the logarithmic forms, known as the laws of logarithms, that follow from the laws of exponents. Let’s recall the laws of logarithms.
Definition: Laws of Logarithms
Suppose , , , and are positive numbers with . The laws of logarithms are
- product: ,
- division: ,
- powers: ,
- change of base: .
These will be useful to solving logarithmic equations with different bases, especially the last one, which allows us to convert a logarithm of one base to another. In order to see where this comes from, first note that which follow directly from the fact that exponents and logarithms are inverses. If we substitute the second expression into the first, we obtain
Using the law of exponents , we can rewrite this as
Taking the logarithm of both sides with base , we find as required. This formula along with the laws of logarithms and the equivalent exponential form allow us to solve logarithmic equations involving logarithms of different bases. Another fact that will be important for solving logarithmic equations is which follows directly from the relationship between logarithmic and exponential forms, in particular, from the fact that the logarithmic and exponential functions are strictly monotonic functions. We can show this directly using the laws of logarithms:
Converting the last expression to exponential form, we find and thus, , as required.
As an example, suppose we want to find the solutions to the logarithmic equation
We first convert the logarithm on the left-hand side to a logarithm of base 2 using the change in base formula: where we have used the fact that to obtain the last line. Thus, using this and the laws of logarithms, the given logarithmic equation becomes
Using the fact that we established, if , then , or converting to exponential form, we obtain
Thus, the only solution to the logarithmic equation is .
Now, let’s consider a few examples to practice and deepen our understanding of solving logarithmic equations. In the first example, we have two logarithms of different bases and the unknown appears inside the logarithm.
Example 1: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers
Find the solution set of in .
Answer
In this example, we want to determine the solution set of a particular logarithmic equation with different bases and the unknown appearing inside the logarithm.
In order to solve the equation, we will make use of the change of base formula, and the power law,
On applying the change of base formula, we can write the right-hand side of the given logarithmic equation, , as a logarithm of base 3 using and :
Using this, the given equation becomes
Since , we have . Thus, the solution set is given by .
Now, let’s consider an example where a logarithmic equation contains two logarithms of different bases and an unknown appearing in each.
Example 2: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers
Determine the solution set of the equation in .
Answer
In this example, we want to determine the solution set of a particular logarithmic equation with different bases and the unknown appearing inside two logarithms of different bases.
In order to solve the equation, we will make use of the change of base formula, and the power law,
On applying the change of base formula, we can write the second term on the left-hand side of the given logarithmic equation, , as a logarithm of base 3 using and :
On substituting this into the given logarithmic equation, we obtain
For and base , , the logarithmic form is equivalent to the exponential form .
Finally, converting to exponential form we have
Thus, the solution set is .
In the next example, we will find the solution to a logarithmic equation that contains the sum of three logarithms of different bases and the unknown appearing inside each of the logarithms.
Example 3: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers
Find the solution set of in .
Answer
In this example, we want to determine the solution set of a particular logarithmic equation with different bases and the unknown appearing inside the logarithm.
In order to solve the equation, we will make use of the change of base formula, and the power law,
On applying the change of base formula, we can write the second term on the left-hand side of the given logarithmic equation, , as a logarithm of base 2 using and , and similarly, for the third term, , using as
Substituting these expressions into the logarithmic equation, we obtain
For and base , , the logarithmic form is equivalent to the exponential form .
Thus, converting to exponential form, we obtain
Therefore, the solution set is .
Now, let’s consider an example where we have to find the solution to a logarithmic equation that contains the sum of the reciprocal of three logarithms of different bases and the unknown appearing inside each of the logarithms.
Example 4: Finding the Solution Set of a Logarithmic Equation over the Set of Real Numbers
Find the solution set of in .
Answer
In this example, we want to determine the solution set of a particular logarithmic equation with different bases and the unknown appearing inside three logarithms of different bases.
In order to solve the equation, we will make use of the change of base formula, and the power law,
On applying the change of base formula, we can write the second term on the left-hand side of the given logarithmic equation, , as a logarithm of base 2 using and , and similarly, for the third term, , using as
Substituting these expressions into the logarithmic equation, we obtain
For and base , , the logarithmic form is equivalent to the exponential form .
Thus, converting to exponential form we find
Thus, the solution set is .
In the next example, we will solve a logarithmic equation that contains logarithms of different bases, including a fractional base, and an unknown appearing inside the logs as a linear and quadratic term.
Example 5: Solving Logarithmic Equations Involving Laws of Logarithms
Find the solution set of in .
Answer
In this example, we want to determine the solution set of a particular logarithmic equation with different bases and the unknown appearing inside two logarithms of different bases.
In order to solve the equation, we will make use of the change of base formula, and the power law,
On applying the change of base formula, we can write the second term on the left-hand side of the given logarithmic equation, , as a logarithm of base 3 using and :
Finally, substituting this expression into the given logarithmic equation, we have
For and base , , the logarithmic form is equivalent to the exponential form .
Finally, converting to exponential form, we have
Thus, the solution set is .
As we have seen in the previous examples, the change of base rule for logarithms also allows us to evaluate expressions of the form by rewriting this logarithm of base as a logarithm of base (i.e., ) along with the power law and the fact that :
Now, let’s consider an example where we have a logarithmic equation that contains a logarithm of the logarithm with two different bases and an unknown appearing inside the logarithm as a quadratic.
Example 6: Solving Logarithmic Equations Involving Laws of Logarithms and Quadratic Equations
Solve , where .
Answer
In this example, we want to determine the solution of a particular logarithmic equation with different bases and an unknown appearing inside the logarithm of a logarithm in the form of a quadratic equation.
For and base , , the exponential form is equivalent to the logarithmic form , which allows us to convert from one form to another once we identify , , and .
Converting to exponential form, we obtain
Repeating the process, we obtain
Therefore, we obtain or . The solution set is .
So far, the examples we have considered had the unknown variable , which we have to solve for, appearing inside the logarithm itself. We can also find solutions to logarithmic equations where the unknown can appear in the base of the logarithm.
In the next example, let’s consider a logarithmic equation with a triple logarithm of different bases and an unknown appearing as the base.
Example 7: Solving Logarithmic Equations over the Set of Real Numbers
Solve , where .
Answer
In this example, we want to determine the solution of a particular logarithmic equation with three different bases and an unknown appearing as a base of a logarithm.
Recall that, for and base , , the logarithmic form is equivalent to the exponential form . The common logarithm where no base is specified has base 10: .
Converting to exponential form, we obtain
Repeating this process for the resulting expression, we get
And repeating it lastly again, we obtain
Thus, the solutions are or , but we ignore the second solution since the logarithm of a negative base (i.e., ), is undefined.
Therefore, the solution to the logarithmic equation is .
Now, let’s consider an example where we have a logarithmic equation containing logarithms of different bases and an unknown appearing both inside the logarithm and the base of the same logarithm.
Example 8: Finding the Solution Set of Exponential Equations Involving Logarithms over the Set of Real Numbers
Find the solution set of in .
Answer
In this example, we want to determine the solution of a particular logarithmic equation with two different bases and an unknown appearing inside and appearing as a base of a logarithm.
In order to solve the given equation, we will make use of the power law:
On applying this and using , we obtain and
Substituting these into the given equation, we obtain
The solutions are and ; however, we will ignore the second as the logarithm of a negative number or with a negative base is undefined. The solution set is therefore .
Using the change of base formula, we can interchange the argument of the logarithm and the base. Changing the logarithm to base and using , we have
Suppose we want to find the solution of ; we want to determine the values of that satisfy this logarithmic equation. Since , this simplifies to
Thus, converting this to exponential form with base , we obtain .
Finally, let’s consider an example where the unknown that we have to solve for appears both inside the logarithm and as a base of another logarithm.
Example 9: Finding the Solution Set of Logarithmic Equations over the Set of Real Numbers
Determine the solution set of the equation in .
Answer
In this example, we want to determine the solution of a particular logarithmic equation with two different bases and an unknown appearing inside and appearing as a base of a logarithm.
In order to solve the logarithmic equation, we will make use of the change in base formula:
If we use the base , this formula allows us to interchange the argument of the logarithm and the base: where we used the fact that . Using this, the given equation becomes
So, if we let , then we have to solve
Multiplying both sides of this equation by and rearranging, we obtain
Thus,
For and base , , the logarithmic form is equivalent to the exponential form .
Thus, upon converting to exponential form, we have
Thus, the solution set is .
Let’s summarize what has been learned in this explainer.
Key Points
- In order to solve the logarithmic equations, we made use of the laws of
logarithms:
- product: ,
- division: ,
- powers: ,
- change of base: .
- If we have a logarithm of base , we can also convert this to a logarithm of base using
- We can also interchange the base and the argument of a logarithm using
- We can also solve logarithmic equations by converting the logarithmic form to the exponential form .
