Exponential EquationsDefinition, 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 children, but with a some of instruction and practice, exponential equations can be solved simply.
This article post will talk about the explanation of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The first step to solving an exponential equation is knowing when you are working with one.
Definition
Exponential equations are equations that have 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 key items to keep in mind for when trying to determine 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 (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you must note is that the variable, x, is in an exponent. The second thing you must not is that there is another term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Once again, the first thing you should notice is that the variable, x, is an exponent. The second thing you must observe is that there are no other value that consists of any variable in them. This implies 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 other functions.
Exponential equations are essential in math and perform a critical role in working out many mathematical questions. Hence, it is important to completely understand what exponential equations are and how they can be used as you go ahead in mathematics.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly common in daily life. There are three primary 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 easily set the two equations equal to each other and figure out for the unknown variable.
2) Equations with different bases on both sides, but they can be made the same using properties 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 case.
3) Equations with different bases on both sides that is impossible to be made the similar. These are the toughest to work out, but it’s possible using the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two latest equations equal to each other and work on the unknown variable. This blog do not cover logarithm solutions, but we will let you know where to get assistance at the very last of this blog.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now learn to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we need to ensue to work on exponential equations.
First, we must recognize 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 through standard algebraic methods.
Third, we have to figure out the unknown variable. Now that we have figured out the variable, we can plug this value back into our first equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to observe how these process work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that both bases are the same. Thus, all you need to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
Now, we change the value of y in the respective equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complex problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. But, both sides are powers of two. In essence, 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 work on this expression to conclude the ultimate result:
28=22x-10
Apply algebra to solve for 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 first equation.
256=49−5=44
Keep seeking for examples and problems on the internet, and if you utilize the properties of exponents, you will inturn master of these theorems, working out most exponential equations without issue.
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