Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a terrifying topic for children, but with a bit of instruction and practice, exponential equations can be determited easily.
This blog post will talk about the explanation of exponential equations, types of exponential equations, steps to work out exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The first step to solving an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that include 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 look for when trying to establish if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is no other term that has the variable in it (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you should notice is that the variable, x, is in an exponent. The second thing you should not is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the contrary, look at this equation:
y = 2x + 5
Once again, the first thing you should observe is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more terms that consists of any variable in them. This means that this equation IS exponential.
You will come across exponential equations when you try solving various calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in arithmetic and perform a pivotal responsibility in solving many math questions. Therefore, it is important to fully grasp what exponential equations are and how they can be used as you go ahead in your math studies.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three primary types of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the most convenient to solve, as we can simply set the two equations same as each other and figure out for the unknown variable.
2) Equations with different bases on each sides, but they can be made the same using rules of the exponents. We will put a few examples below, but by making the bases the same, you can observe the exact steps as the first event.
3) Equations with distinct bases on each sides that is unable to be made the same. These are the most difficult to figure out, but it’s possible using the property of the product rule. By increasing two or more factors to identical 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 do not cover logarithm solutions, but we will tell you where to get guidance at the end of this article.
How to Solve Exponential Equations
From 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
We have three steps that we are required to follow to solve exponential equations.
First, we must determine the base and exponent variables within the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Thereafter, we can solve them through standard algebraic methods.
Third, we have to figure out the unknown variable. Since we have solved for the variable, we can plug this value back into our initial equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to observe how these procedures work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Hence, all you are required to do is to restate the exponents and figure them out utilizing algebra:
y+1=3y
y=½
Now, we replace the value of y in the respective equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complicated question. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a identical base. Despite that, both sides are powers of two. In essence, the working comprises of decomposing 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 conclude the ultimate answer:
28=22x-10
Apply algebra to figure out x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our work by substituting 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and questions over the internet, and if you use the rules of exponents, you will inturn master of these concepts, figuring out almost all exponential equations without issue.
Better Your Algebra Abilities with Grade Potential
Solving questions with exponential equations can be tough with lack of help. Although this guide take you through the essentials, you still may find questions or word problems that make you stumble. Or maybe you need some extra guidance as logarithms come into play.
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