Exponential EquationsDefinition, Workings, and Examples
In arithmetic, an exponential equation occurs when the variable shows up in the exponential function. This can be a terrifying topic for kids, but with a some of direction and practice, exponential equations can be determited simply.
This article post will discuss the explanation of exponential equations, types of exponential equations, steps to figure out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that have the variable in an exponent. For example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two major things to bear in mind for when attempting to figure out 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 (aside from the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you must note is that the variable, x, is in an exponent. Thereafter thing you must observe is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no more terms that consists of any variable in them. This means that this equation IS exponential.
You will run into exponential equations when solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in arithmetic and perform a pivotal duty in working out many computational questions. Thus, it is critical to fully understand what exponential equations are and how they can be used as you move ahead in arithmetic.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in daily life. There are three main types of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be made the same using properties of the exponents. We will show some examples below, but by changing the bases the equal, you can observe the exact steps as the first instance.
3) Equations with variable bases on both sides that cannot be made the same. These are the trickiest to figure out, but it’s feasible using the property of the product rule. By raising both factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can determine the two new equations equal to each other and solve for the unknown variable. This article does not include logarithm solutions, but we will let you know where to get help at the closing parts of this article.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now understand how to solve any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
There are three steps that we are required to ensue to solve exponential equations.
First, we must determine the base and exponent variables in the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Then, we can solve them using standard algebraic rules.
Lastly, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our first equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to observe how these procedures work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Thus, all you have to do is to rewrite the exponents and figure them out through algebra:
y+1=3y
y=½
Now, we change the value of y in the specified equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complex problem. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. In essence, the working consists of decomposing both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the ultimate answer:
28=22x-10
Carry out algebra to solve for x in the exponents as we did in the prior example.
8=2x-10
x=9
We can verify our workings by altering 9 for x in the original equation.
256=49−5=44
Continue seeking for examples and questions over the internet, and if you use the laws of exponents, you will inturn master of these theorems, solving almost all exponential equations with no issue at all.
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