Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation takes place when the variable appears in the exponential function. This can be a frightening topic for kids, but with a some of instruction and practice, exponential equations can be solved quickly.
This blog post will discuss the definition of exponential equations, kinds of exponential equations, proceduce to work out exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is understanding when you have 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 things to bear in mind for when attempting 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 (aside from the exponent)
For example, look at this equation:
y = 3x2 + 7
The first thing you should observe is that the variable, x, is in an exponent. Thereafter thing you must notice is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the flipside, check out this equation:
y = 2x + 5
Yet again, the first thing you must observe is that the variable, x, is an exponent. The second thing you should notice is that there are no other terms that have the variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when working on various calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are essential in arithmetic and perform a pivotal duty in solving many math problems. Therefore, it is critical to completely grasp what exponential equations are and how they can be used as you progress in mathematics.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly ordinary in everyday life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient to work out, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be created similar using properties of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the described steps as the first case.
3) Equations with distinct bases on both sides that is impossible to be made the similar. These are the most difficult to figure out, but it’s attainable utilizing the property of the product rule. By raising both 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 each other and solve for the unknown variable. This article does not include logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now learn to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to solve exponential equations.
Primarily, we must identify the base and exponent variables within the equation.
Next, we are required to rewrite an exponential equation, so all terms are in common base. Thereafter, we can solve them through standard algebraic methods.
Lastly, we have to solve for the unknown variable. Now that we have figured out the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Hence, all you need to do is to rewrite the exponents and solve through algebra:
y+1=3y
y=½
Right away, 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 complex sum. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation does not share a identical base. But, both sides are powers of two. As such, the solution comprises of breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to find the final result:
28=22x-10
Carry out algebra to figure out x in the exponents as we performed in the last example.
8=2x-10
x=9
We can recheck our work by altering 9 for x in the original equation.
256=49−5=44
Keep looking for examples and problems over the internet, and if you use the laws of exponents, you will become a master of these concepts, solving almost all exponential equations without issue.
Improve Your Algebra Skills with Grade Potential
Solving questions with exponential equations can be difficult without help. Although this guide take you through the basics, you still might face questions or word questions that might stumble you. Or possibly you require some further guidance as logarithms come into the scene.
If you feel the same, think about signing up for a tutoring session with Grade Potential. One of our expert teachers can support you improve your abilities and mental state, so you can give your next examination a grade-A effort!
