In this explainer, we will learn how to solve logarithmic equations with like bases using the laws of exponents and logarithms.

A logarithmic equation is an equation with an unknown variable in part of the logarithm, usually the argument. In the case of logarithmic equations with a single logarithm, it is possible to rewrite them in exponential form to make them easier to solve. Let’s recall the definition of a logarithm to show how to do this.

### Definition: Logarithms

For an exponential equation in the form , where , this can be written as the logarithmic equation where is the base of the logarithm, is the argument, and is the exponent.

Therefore, .

We can see from the definition above that if an unknown variable is in the argument, , then by rewriting as we can make the subject of the equation, making it possible to solve for an unknown variable.

In the first example, we will discuss how to find an unknown variable in the argument by rewriting the logarithm in its exponential form, as detailed above.

### Example 1: Solving Logarithmic Equations with a Single Logarithm

Given that , find the value of .

### Answer

As we are given a logarithmic equation with a single logarithm, we can rearrange the logarithm to make the argument, , the subject by rewriting the logarithm as an exponential equation. Recall that where and .

Therefore, for the logarithmic equation , we can rewrite this as

Evaluating 12 to the power of 1, and solving for , we get

Therefore, the value of in the logarithmic equation is 9.

In the next example, we will solve another logarithmic equation with a single logarithm by rewriting it as an exponential equation, but this time with an unknown in the base.

### Example 2: Solving a Logarithmic Equation with a Single Logarithm

What is the solution set of the equation ?

### Answer

As we are given a logarithmic equation with a single logarithm, with an unknown variable in the base, by rewriting this as an exponential equation, we can more easily solve it. Recall that where and .

Therefore, we can rearrange by writing it as the exponential equation

We can solve this by taking the square root of both sides of the equation, giving us

As the base, , must be greater than zero, is the only possible solution, so we discard . Solving for , we then get

Therefore, the solution set for is .

So far, we have considered logarithmic equations with a single logarithm. Next, we will discuss how to solve logarithmic equations with multiple logarithms of the same base.

Let’s recall the laws of logarithms for like bases and special values.

### Law: Laws of Logarithms for Like Bases and Special Values

For a logarithm with base , where ,

- ;
- ;
- multiplication law: where and ;
- division law: where and ;
- power law: where .

Where we have like bases, we can use the laws of logarithms to first combine logarithms, and then solve by either rearranging or making the arguments equal, depending on the equation.

We will demonstrate how to use the laws of logarithms for like bases to solve a logarithmic equation in our next example.

### Example 3: Finding the Solution Set for a Logarithmic Equation with Like Bases

Determine the solution set of the equation in .

### Answer

In the logarithmic equation , as all parts of the equation have logarithms with the same base, 8, we can apply the laws of logarithms for like bases in order to simplify and solve.

Since we are adding two logarithms on the left-hand side of the equation, we can use the law for multiplication of logarithms to simplify this. The law states that where , , and .

Applying this to the left-hand side, we get

Now, since both the left-hand side and right-hand side of the equation are written as a single logarithm with the same base, therefore the arguments must be equal, giving us

Expanding the brackets and rearranging to equal zero, we get

Factoring and solving, we then get

Now, since the argument of a logarithm must be greater than zero, both and must be greater than zero. As such, is not a valid solution, so must equal 10.

Therefore, the solution set of the equation is .

In the next example, we will consider how to use multiple laws of logarithms with like bases to solve a logarithmic equation.

### Example 4: Finding the Solution Set for a Logarithmic Equation with Like Bases

Find the solution set of in .

### Answer

In the logarithmic equation , as all parts of the equation have logarithms with the same base, 2, we can apply the laws of logarithms for like bases in order to simplify and solve.

Since we have a multiplication of a number by a logarithm in the first term, we need to use the power law for logarithms, which states that where and .

Therefore, , giving us

Next, as we are subtracting one logarithm from another with the same base on the left-hand side of the equation, we can simplify this using the division law for logarithms, which states that where , , and .

Applying this law, we get

Now, since both the left-hand side and the right-hand side of the equation are written as a single logarithm with the same base, the arguments must be equal, giving us

As both the numerator and the denominator of the left-hand side are a power of 4, we can take the positive and negative fourth root, giving us

Solving for , we get

Solving for , we get

Therefore, the solution set of is .

In our last example, we will discuss how to use the laws of logarithms to solve a geometric problem.

### Example 5: Using Laws of Logarithms to Solve a Geometric Problem

Given that and , determine the value of . Give your answer to the nearest tenth.

### Answer

As seen in the figure, we have one large right triangle, , with side and hypotenuse , with and given, as well as a smaller right triangle, , inside the larger one, with side and hypotenuse given. As both triangles are right and share a common angle, , they must be similar triangles. Therefore, the ratios of the sides are equal, giving us which is the same as

We can see from the diagram that and

Therefore, substituting into , we get

As we are working with lengths of sides of triangles, must be positive, so

To find , we can rewrite this as an exponential equation since where and .

Therefore,

To solve for , we recall that if , then , giving us

Therefore, is 1.8 to the nearest tenth.

In this explainer, we have discussed how to solve logarithmic equations using their exponential forms, and using laws of logarithms for equations with like bases. Let’s recap the key points.

### Key Points

- A logarithmic equation is an equation with an unknown in part of the logarithm.
- If a logarithmic equation has an unknown in its base or argument and contains a single logarithm, then we can rearrange it using its exponential form.
- If a logarithmic equation contains multiple logarithms with the same base, then we can use the laws of logarithms to simplify it, and then either equate arguments if there are single logarithms on both sides or write it in its exponential form if there is a logarithm on only one side.