Exponential and Logarithmic Equations - Revision Notes

In these revision notes for Exponential and Logarithmic Equations, we cover the following key points:

• What are exponential equations?
• How many types of exponential equations are there?
• How do we solve the exponential equation without involving logarithms?
• How do we solve the exponential equation by involving logarithms?
• What are logarithmic equations?
• How many types of logarithmic equations are there?
• How do we solve standard logarithmic equations?
• How do we solve more advanced logarithmic equations?

Exponential and Logarithmic Equations Revision Notes

Exponential equations are those equations that have the variable(s) in the exponent. They have different methods of solution, depending on the complexity of the parts involved.

1. Exponential equations with a single exponential term

The simplest case of exponential equations involves situations where there is a single exponential term on one side and the result on the other side of the equation, which is a power of the base number that contains the variable in the exponent. The general form of such equations is

The first thing to check when trying to solve such equations is to try writing the result as an exponential number with the same base as the term that contains the variable. If you are able to do this, then you can forget the common base and focus only on the exponential part, where you obtain a type of known equations (usually linear).

2. Exponential equations with more than one exponential term but where all terms can be written at the same base

If the exponential equation contains more than one exponential part, the solution procedure may require some kind of factorisation. However, if it is possible to write all terms at the same base, this is not a problem.

3. Exponential equations where the variable appears in more than one term

If you are able to express every term to the same base, it is still possible to solve such exponential equations through the standard methods (i.e. without involving logarithms).

4. Solving exponential equations where the terms cannot turn to a common base

If the terms of an exponential equation are impossible to write at the same base, we must use the logarithmic approach to solve them. Obviously, the base of the logarithm on both sides must be equal to the argument of the term that contains the variable. Only in this way, we can eliminate the exponent from the equation. The simplest case would be when there is a single exponential term and a constant on the other side of the equation but the logarithmic method can be extended further to include a wide range of exponential equations.

5. Exponential equations where the variable is written to the second power

Sometimes, the variable of an exponential equation is written in the second power. In such cases, we have to solve a quadratic equation when turning all terms at the same base is impossible; otherwise, we use the logarithmic approach.

Logarithmic equations are those equations that contain the variable in the argument of a logarithm. There are three types of logarithmic equations, as listed below.

1. Equations with a single logarithmic term

In these equations, we send the variable out of the logarithm by using the logarithmic to exponential conversion. The general form of such equations is

where x is the variable.

2. Equations with more than one logarithmic term but with the same base

In these equations, we have to focus only on the arguments to solve them. This means to disregard the bases during the solution and consider only the arguments. However, this is done only after having expressed the equation to a single logarithmic term in either side.

3. Equations with more than one logarithmic term with different bases

In these equations, the first thing to do is to attempt to write the logarithms with the same base. If this is possible, the equation is solved by using the second method described above; otherwise, the result is usually obtained by applying the logarithmic property

Obviously, there are more challenging exponential and logarithmic equations available in various textbooks, depending on the degree of difficulty they have.