Exponential EquationsExplanation, Workings, and Examples
In arithmetic, an exponential equation occurs when the variable shows up in the exponential function. This can be a frightening topic for kids, but with a some of direction and practice, exponential equations can be solved simply.
This blog post will talk about the explanation of exponential equations, kinds of exponential equations, process to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The initial step to work on an exponential equation is determining 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 key items to look 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 (aside from the exponent)
For example, look at this equation:
y = 3x2 + 7
The primary thing you must observe 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 signifies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
One more time, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no other value that includes any variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when solving various calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and play a central role in working out many mathematical problems. Hence, it is crucial to fully grasp what exponential equations are and how they can be used as you move ahead in your math studies.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in daily life. There are three major kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the easiest to work out, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with different bases on each sides, but they can be made similar utilizing properties of the exponents. We will take a look at some examples below, but by making the bases the equal, you can follow the exact steps as the first case.
3) Equations with distinct bases on each sides that is unable to be made the similar. These are the most difficult to solve, but it’s attainable using the property of the product rule. By increasing two or more factors to the same power, we can multiply the factors on each side and raise them.
Once we have done this, we can set the two new equations equal to each other and figure out the unknown variable. This article do not include logarithm solutions, but we will let you know where to get help at the end of this blog.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now move on to how to work on any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are required to ensue to work on exponential equations.
Primarily, we must recognize the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them utilizing standard algebraic techniques.
Third, we have to solve for the unknown variable. Since 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 Work on Exponential Equations
Let's check out some examples to note how these procedures work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are identical. Therefore, all you have to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we replace the value of y in the specified equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complicated question. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. However, both sides are powers of two. By itself, the solution comprises of breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to find the ultimate result:
28=22x-10
Carry out algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can recheck our work by substituting 9 for x in the first equation.
256=49−5=44
Keep searching for examples and questions over the internet, and if you use the rules of exponents, you will become a master of these concepts, working out almost all exponential equations with no issue at all.
Better Your Algebra Skills with Grade Potential
Solving problems with exponential equations can be tricky with lack of help. While this guide take you through the essentials, you still may face questions or word questions that make you stumble. Or possibly you require some additional assistance as logarithms come into the scenario.
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