Exponential EquationsDefinition, Workings, and Examples
In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a frightening topic for students, but with a bit of instruction and practice, exponential equations can be solved simply.
This article post will discuss the definition of exponential equations, kinds of exponential equations, steps to work out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The primary step to solving an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that include 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 primary things to keep 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, take a look at this equation:
y = 3x2 + 7
The primary thing you must notice is that the variable, x, is in an exponent. The second thing you must notice is that there is another term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
One more time, the first 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 includes any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when you try solving diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are essential in math and play a central responsibility in working out many mathematical questions. Thus, it is crucial to fully grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in daily life. There are three main kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the most convenient to work out, 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 both sides, but they can be made 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 instance.
3) Equations with different bases on both sides that is unable to be made the same. These are the most difficult to figure out, but it’s attainable using the property of the product rule. By increasing both factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can set the two latest equations equal to one another and figure out 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 blog.
How to Solve Exponential Equations
Knowing the explanation and kinds of exponential equations, we can now move on to how to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to work on exponential equations.
Primarily, we must recognize the base and exponent variables inside the equation.
Next, 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. Since we have figured out the variable, we can plug this value back into our original equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to see how these procedures work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can see that both bases are identical. Therefore, all you need to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we substitute the value of y in the respective 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 identical base. Despite that, both sides are powers of two. In essence, the working consists of decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the ultimate result:
28=22x-10
Carry out algebra to figure out 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
Continue looking for examples and questions on the internet, and if you utilize the laws of exponents, you will inturn master of these concepts, working out almost all exponential equations with no issue at all.
Improve Your Algebra Abilities with Grade Potential
Solving questions with exponential equations can be tough without guidance. Although this guide take you through the fundamentals, you still may find questions or word questions that may hinder you. Or maybe you desire some additional help as logarithms come into the scenario.
If this is you, consider signing up for a tutoring session with Grade Potential. One of our expert teachers can support you enhance your abilities and confidence, so you can give your next test a grade-A effort!
