Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for students, but with a some of instruction and practice, exponential equations can be solved easily.
This article post will discuss the explanation of exponential equations, kinds of exponential equations, proceduce to work out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that consist of 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 major things to keep in mind for when you seek to figure out if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is no other term that has the variable in it (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you must observe is that the variable, x, is in an exponent. The second thing you must not is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the primary thing you should notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no other terms that have the variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when solving various calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are crucial in math and play a central responsibility in working out many math problems. Thus, it is critical to completely grasp what exponential equations are and how they can be utilized as you move ahead in mathematics.
Types of Exponential Equations
Variables occur 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 work out:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can simply set the two equations same as each other and work out for the unknown variable.
2) Equations with different bases on each sides, but they can be created the same using rules of the exponents. We will put a few examples below, but by converting the bases the same, you can follow the same steps as the first event.
3) Equations with variable bases on each sides that is impossible to be made the similar. These are the trickiest to figure out, but it’s possible using the property of the product rule. By increasing two or more 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 identical to each other and figure out the unknown variable. This blog does not cover logarithm solutions, but we will let you know where to get assistance at the very last of this article.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now understand how to solve any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
Remember these three steps that we are going to follow to work on exponential equations.
Primarily, we must determine the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Then, we can solve them utilizing standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Once 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 take a loot at a few examples to observe how these steps work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are identical. Thus, all you need to do is to restate the exponents and figure them out utilizing algebra:
y+1=3y
y=½
So, we substitute 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 more complex question. Let's solve 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 breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to find the final result:
28=22x-10
Perform algebra to work out the x in the exponents as we performed in the last example.
8=2x-10
x=9
We can double-check our workings by replacing 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and questions on the internet, and if you use the properties of exponents, you will become a master of these concepts, working out most exponential equations without issue.
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Solving questions with exponential equations can be tough with lack of support. Although this guide covers the fundamentals, you still might encounter questions or word questions that make you stumble. Or perhaps you require some extra help as logarithms come into play.
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