Exponential EquationsExplanation, Workings, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a frightening topic for children, but with a bit of direction and practice, exponential equations can be solved simply.
This blog post will talk about the explanation of exponential equations, types of exponential equations, proceduce 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 are working with 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 primary items to look for when attempting to determine if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is no other term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. Thereafter thing you must observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
Once again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you must note is that there are no more value that includes any variable in them. This means that this equation IS exponential.
You will come across exponential equations when you try solving various calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are very important in mathematics and play a pivotal duty in figuring out many math questions. Therefore, it is crucial to completely grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly ordinary in everyday life. There are three major kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can simply set the two equations equal to each other and figure out for the unknown variable.
2) Equations with distinct bases on both sides, but they can be created similar employing rules of the exponents. We will show some examples below, but by changing the bases the same, you can observe the exact steps as the first case.
3) Equations with distinct bases on each sides that is impossible to be made the same. These are the most difficult to figure out, but it’s feasible using the property of the product rule. By raising two or more factors to the same power, we can multiply the factors on each side and raise them.
Once we are done, we can resolute the two latest equations identical to each other and figure out the unknown variable. This article does not cover logarithm solutions, but we will let you know where to get assistance at the end of this article.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now learn to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we are going to follow to work on exponential equations.
First, we must identify the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them using standard algebraic techniques.
Third, we have to work on the unknown variable. Once we have solved for the variable, we can put this value back into our initial equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to observe how these process work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can see that both bases are identical. Thus, all you need to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
So, we replace the value of y in the given equation to corroborate that the form is true:
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 do not share a common base. But, both sides are powers of two. In essence, the solution consists of breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the final result:
28=22x-10
Apply algebra to work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our answer by replacing 9 for x in the first equation.
256=49−5=44
Continue searching for examples and problems on the internet, and if you use the rules of exponents, you will become a master of these theorems, solving almost all exponential equations without issue.
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Working on questions with exponential equations can be tricky without guidance. Although this guide goes through the essentials, you still may face questions or word questions that make you stumble. Or maybe you require some extra guidance as logarithms come into the scene.
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