Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation takes place when the variable shows up in the exponential function. This can be a scary topic for students, but with a some of instruction and practice, exponential equations can be determited simply.
This article post will discuss the definition of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The primary step to work on an exponential equation is understanding when you have one.
Definition
Exponential equations are equations that include 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 keep in mind 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 no other term that has the variable in it (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The primary thing you should note is that the variable, x, is in an exponent. The second thing you must observe is that there is another term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the primary thing you must notice is that the variable, x, is an exponent. The second thing you should observe is that there are no more value that consists of any variable in them. This means that this equation IS exponential.
You will come across exponential equations when working on diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in math and perform a pivotal duty in figuring out many math questions. Therefore, it is crucial to fully grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three primary kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the easiest to work out, as we can simply set the two equations equal to each other and solve for the unknown variable.
2) Equations with distinct bases on both sides, but they can be created the same utilizing properties of the exponents. We will show some examples below, but by changing the bases the same, you can follow the described steps as the first case.
3) Equations with different bases on each sides that cannot be made the same. These are the toughest to solve, but it’s attainable through the property of the product rule. By raising both factors to similar power, we can multiply the factors on each side and raise them.
Once we are done, we can set the two latest equations equal to each other and solve for the unknown variable. This blog do not contain logarithm solutions, but we will let you know where to get help at the closing parts of this article.
How to Solve Exponential Equations
After going through the definition and types of exponential equations, we can now understand how to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to solve exponential equations.
Primarily, we must identify the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them using standard algebraic rules.
Third, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our original equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these process work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Hence, all you need to do is to restate the exponents and work on them through algebra:
y+1=3y
y=½
Right away, we replace 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 further complicated sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a similar base. But, both sides are powers of two. In essence, the solution comprises of decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final result:
28=22x-10
Apply algebra to figure out x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can verify our workings by altering 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems over the internet, and if you use the rules of exponents, you will inturn master of these concepts, working out almost all exponential equations with no issue at all.
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Working on questions with exponential equations can be difficult without guidance. Although this guide goes through the fundamentals, you still might find questions or word problems that make you stumble. Or perhaps you need some additional assistance as logarithms come into play.
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