Exponential EquationsDefinition, Workings, and Examples
In math, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for children, but with a some of direction and practice, exponential equations can be worked out quickly.
This article post will talk about the definition of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to solving an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that have 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 items to look for when trying to figure out 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 (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you should observe is that the variable, x, is in an exponent. The second thing you should notice is that there is another term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the contrary, 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 must note is that there are no more value that consists of any 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 various distinct functions.
Exponential equations are crucial in mathematics and perform a critical role in solving many computational questions. Hence, it is important to fully grasp what exponential equations are and how they can be utilized as you move ahead in your math studies.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in everyday life. There are three main kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the simplest to work out, as we can easily set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same utilizing rules of the exponents. We will show some examples below, but by converting the bases the equal, you can observe the described steps as the first instance.
3) Equations with variable bases on each sides that cannot be made the similar. These are the most difficult to solve, but it’s feasible through the property of the product rule. By raising two or more 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 equal to one another and work on the unknown variable. This blog do not cover logarithm solutions, but we will tell you where to get guidance at the very last of this article.
How to Solve Exponential Equations
From the explanation and types of exponential equations, we can now understand how to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to solve exponential equations.
First, we must recognize the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Then, we can solve them using standard algebraic techniques.
Lastly, we have to figure out the unknown variable. Since we have figured out the variable, we can put this value back into our first equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's look at some examples to note how these steps work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are identical. Therefore, all you are required 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 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. However, both sides are powers of two. In essence, the working includes decomposing respectively 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 final result:
28=22x-10
Carry out algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can double-check our workings by altering 9 for x in the first equation.
256=49−5=44
Keep seeking for examples and problems over the internet, and if you use the laws of exponents, you will inturn master of these concepts, figuring out most exponential equations without issue.
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Solving problems with exponential equations can be difficult without help. While this guide covers the fundamentals, you still may find questions or word problems that make you stumble. Or perhaps you require some extra help as logarithms come into play.
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