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October 18, 2022

Exponential EquationsDefinition, Workings, and Examples

In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a terrifying topic for students, but with a bit of direction and practice, exponential equations can be solved easily.

This blog post will discuss the explanation of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's began!

What Is an Exponential Equation?

The first step to solving an exponential equation is determining when you have one.

Definition

Exponential equations are equations that include the variable in an exponent. For instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.

There are two major items to look for when attempting to establish if an equation is exponential:

1. The variable is in an exponent (meaning it is raised to a power)

2. There is only one term that has the variable in it (in addition of the exponent)

For example, take a look at this equation:

y = 3x2 + 7

The most important thing you should observe is that the variable, x, is in an exponent. Thereafter thing you must notice is that there is another term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.

On the other hand, take a look at this equation:

y = 2x + 5

One more time, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more terms that consists of any variable in them. This means that this equation IS exponential.

You will come upon exponential equations when you try solving various calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.

Exponential equations are crucial in mathematics and perform a central duty in solving many math questions. Therefore, it is important to completely grasp what exponential equations are and how they can be utilized as you progress in arithmetic.

Types of Exponential Equations

Variables appear in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three primary types of exponential equations that we can work out:

1) Equations with identical bases on both sides. This is the easiest to solve, as we can easily set the two equations equivalent as each other and figure out for the unknown variable.

2) Equations with dissimilar bases on both sides, but they can be created the same using properties of the exponents. We will take a look at some examples below, but by converting the bases the equal, you can observe the same steps as the first case.

3) Equations with distinct bases on both sides that is impossible to be made the similar. These are the toughest to work out, but it’s attainable through the property of the product rule. By raising both factors to identical power, we can multiply the factors on each side and raise them.

Once we are done, we can set the two new equations equal to each other and solve for the unknown variable. This article does not include logarithm solutions, but we will tell you where to get guidance at the end of this blog.

How to Solve Exponential Equations

After going through the definition and kinds of exponential equations, we can now move on to how to solve any equation by ensuing these easy procedures.

Steps for Solving Exponential Equations

We have three steps that we are going to follow to solve exponential equations.

First, we must identify the base and exponent variables in the equation.

Next, we are required to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them using standard algebraic rules.

Third, we have to work on the unknown variable. Since we have figured out the variable, we can put this value back into our initial equation to discover the value of the other.

Examples of How to Work on Exponential Equations

Let's take a loot at some examples to note how these steps work in practice.

First, we will work on the following example:

7y + 1 = 73y

We can observe that both bases are the same. Hence, all you need to do is to restate the exponents and work on them using algebra:

y+1=3y

y=½

Now, we substitute the value of y in the given equation to corroborate that the form is real:

71/2 + 1 = 73(½)

73/2=73/2

Let's observe this up with a further complicated sum. Let's solve this expression:

256=4x−5

As you have noticed, the sides of the equation does not share a identical base. But, both sides are powers of two. As such, the solution includes breaking down both the 4 and the 256, and we can replace the terms as follows:

28=22(x-5)

Now we figure out this expression to come to the ultimate answer:

28=22x-10

Apply algebra to solve for x in the exponents as we performed in the last example.

8=2x-10

x=9

We can verify our answer by replacing 9 for x in the initial equation.

256=49−5=44

Keep looking for examples and questions online, and if you use the laws of exponents, you will become a master of these theorems, figuring out almost all exponential equations without issue.