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 instruction and practice, exponential equations can be solved easily.
This article post will talk about the definition of exponential equations, types of exponential equations, process to figure out exponential equations, and examples with answers. Let's began!
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 instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary items to bear in mind for when you seek 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, look at this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. The second thing you should observe is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the contrary, 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 must notice 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 diverse calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are crucial in arithmetic and play a pivotal role in solving many math problems. Therefore, it is critical to fully understand what exponential equations are and how they can be utilized as you move 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 daily life. There are three major kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can easily set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same using rules 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 instance.
3) Equations with different bases on each sides that cannot be made the similar. These are the most difficult to figure out, but it’s possible utilizing 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 have done this, we can resolute the two latest equations equal to one another and work on the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now understand how to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we are required to follow to solve exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them utilizing standard algebraic methods.
Third, we have to solve for the unknown variable. Now that we have figured out the variable, we can put this value back into our initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's check out a few examples to see how these process work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Hence, all you are required 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 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 complex question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a similar base. But, both sides are powers of two. As such, the working comprises of breaking down both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we figure out this expression to find the ultimate answer:
28=22x-10
Apply algebra to solve for x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can double-check our work by altering 9 for x in the initial equation.
256=49−5=44
Keep seeking for examples and problems on the internet, and if you utilize the laws of exponents, you will inturn master of these concepts, figuring out most exponential equations with no issue at all.
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Working on questions with exponential equations can be difficult with lack of help. Although this guide take you through the basics, you still might encounter questions or word questions that make you stumble. Or perhaps you require some further guidance as logarithms come into the scene.
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