Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a terrifying topic for students, but with a some of direction and practice, exponential equations can be worked out quickly.
This blog post will talk about the definition of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The first step to work on an exponential equation is knowing 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 key things to look for when you seek 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, check out this equation:
y = 3x2 + 7
The most important thing you must note 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 means that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you should note is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more value that consists of any variable in them. This means that this equation IS exponential.
You will come across exponential equations when working on various calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are crucial in mathematics and play a central role in solving many computational questions. Thus, it is critical to completely understand what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly easy to find in everyday 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 simplest to work out, as we can easily set the two equations equal to each other and work 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 show some examples below, but by changing the bases the same, you can observe the same steps as the first instance.
3) Equations with different bases on both sides that cannot be made the similar. These are the trickiest to work out, but it’s possible through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two latest equations identical to one another and solve for the unknown variable. This article do not contain logarithm solutions, but we will let you know where to get assistance at the very last of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now learn to work on any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we need 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 are in common base. Then, we can solve them through standard algebraic rules.
Third, we have to solve for the unknown variable. Since we have figured out the variable, we can plug this value back into our original equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to observe how these process work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are identical. Hence, all you need to do is to restate the exponents and solve through algebra:
y+1=3y
y=½
Now, we substitute the value of y in the given equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complicated question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. However, both sides are powers of two. By itself, the solution consists of decomposing respectively 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 final result:
28=22x-10
Apply algebra to work out the x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can double-check our workings by substituting 9 for x in the original equation.
256=49−5=44
Keep searching for examples and problems over the internet, and if you utilize the rules of exponents, you will inturn master of these concepts, solving almost all exponential equations without issue.
Enhance Your Algebra Skills with Grade Potential
Solving problems with exponential equations can be tough with lack of guidance. Even though this guide take you through the basics, you still might encounter questions or word problems that make you stumble. Or maybe you desire some additional help as logarithms come into the scene.
If this is you, contemplate signing up for a tutoring session with Grade Potential. One of our experienced instructors can help you better your abilities and confidence, so you can give your next exam a grade-A effort!
