Exponential EquationsDefinition, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a scary 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, proceduce to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is understanding when you have 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 things to keep in mind for when attempting to determine 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 (besides the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you must observe is that the variable, x, is in an exponent. Thereafter thing you must observe is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Once again, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when you try solving various calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in mathematics and play a pivotal responsibility in working out many computational questions. Therefore, it is important to completely understand what exponential equations are and how they can be used as you move ahead in your math studies.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable easy to find in daily life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made the same using rules of the exponents. We will take a look at some examples below, but by changing the bases the equal, you can observe the described steps as the first event.
3) Equations with different bases on each sides that cannot be made the similar. These are the trickiest to solve, but it’s possible using the property of the product rule. By increasing both 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 latest equations identical to each other and solve for the unknown variable. This blog do not contain logarithm solutions, but we will tell you where to get guidance at the very last of this article.
How to Solve Exponential Equations
Knowing the explanation and kinds of exponential equations, we can now learn to solve any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
We have three steps that we are required to follow to work on exponential equations.
First, we must determine the base and exponent variables in 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.
Lastly, we have to solve for the unknown variable. Now that we have solved for the variable, we can plug this value back into our original equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at some examples to note how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Thus, all you are required to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
Now, we substitute 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 further complicated problem. Let's solve 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 solution includes decomposing both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to come to the ultimate answer:
28=22x-10
Apply algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can recheck our answer by replacing 9 for x in the initial equation.
256=49−5=44
Keep searching for examples and problems over the internet, and if you use the properties of exponents, you will turn into a master of these theorems, solving most exponential equations without issue.
Better Your Algebra Skills with Grade Potential
Solving questions with exponential equations can be tough with lack of guidance. Even though this guide covers the fundamentals, you still may encounter questions or word questions that might stumble you. Or perhaps you need some further guidance as logarithms come into play.
If you feel the same, consider signing up for a tutoring session with Grade Potential. One of our professional tutors can support you enhance your skills and mental state, so you can give your next exam a grade-A effort!
