Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation arises when the variable shows up in the exponential function. This can be a terrifying topic for students, but with a bit of instruction and practice, exponential equations can be worked out quickly.
This article post will talk about the definition of exponential equations, types of exponential equations, process to figure out exponential equations, and examples with solutions. 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 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 major items to keep in mind for when you seek 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 (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you should notice is that the variable, x, is in an exponent. The second thing you should notice is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
One more time, the first thing you should observe is that the variable, x, is an exponent. Thereafter thing you should 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 solving various calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are crucial in mathematics and perform a pivotal responsibility in solving many mathematical questions. Thus, it is important to fully grasp what exponential equations are and how they can be used as you go ahead in arithmetic.
Kinds of Exponential Equations
Variables appear 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 work out:
1) Equations with identical bases on both sides. This is the easiest to work out, as we can easily set the two equations equivalent as each other and figure 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 take a look at some examples below, but by changing the bases the same, you can observe the same steps as the first event.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the toughest to work out, but it’s possible utilizing the property of the product rule. By raising two or more factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can set the two latest equations equal to one another and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get assistance at the closing parts of this article.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now learn 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.
First, we must identify the base and exponent variables within the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them utilizing standard algebraic rules.
Third, we have to solve for the unknown variable. Since we have figured out the variable, we can plug this value back into our first equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's check out some examples to observe how these steps work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are identical. Thus, all you have to do is to rewrite the exponents and solve utilizing algebra:
y+1=3y
y=½
Right away, we replace the value of y in the specified equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complex sum. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does 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 come to the final result:
28=22x-10
Apply algebra to figure out x in the exponents as we did in the previous example.
8=2x-10
x=9
We can double-check our workings by substituting 9 for x in the initial equation.
256=49−5=44
Continue seeking for examples and problems online, and if you use the laws of exponents, you will inturn master of these concepts, working out almost all exponential equations without issue.
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