Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation arises when the variable shows up in the exponential function. This can be a scary topic for children, but with a some of direction and practice, exponential equations can be determited easily.
This article post will discuss the explanation of exponential equations, types of exponential equations, steps to solve exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that consist of 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 key things to look for when trying to figure out if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is only one term that has the variable in it (aside from the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you should notice is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Once again, the first thing you should notice is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no other value 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 essential in math and play a central duty in figuring out many computational questions. Hence, it is critical to fully grasp what exponential equations are and how they can be used as you progress in mathematics.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three major types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the easiest to solve, as we can simply set the two equations equal to each other and solve for the unknown variable.
2) Equations with different bases on each sides, but they can be created the same using rules of the exponents. We will take a look at some examples below, but by making the bases the same, you can observe the same steps as the first instance.
3) Equations with different bases on both sides that is unable to be made the similar. These are the most difficult to solve, but it’s possible through the property of the product rule. By increasing both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two latest equations equal to one another and figure out the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get help at the very last of this blog.
How to Solve Exponential Equations
From the explanation and kinds of exponential equations, we can now learn to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we need to ensue to solve exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them through standard algebraic techniques.
Third, 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 check out a few examples to observe how these procedures work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Therefore, all you have to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we substitute the value of y in the respective equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex question. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation does not share a identical base. Despite that, both sides are powers of two. By itself, the solution consists of decomposing both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the ultimate result:
28=22x-10
Carry out algebra to work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can recheck our workings by altering 9 for x in the first equation.
256=49−5=44
Keep looking for examples and questions over the internet, and if you use the laws of exponents, you will inturn master of these theorems, working out almost all exponential equations without issue.
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