Exponential EquationsDefinition, Workings, and Examples
In math, an exponential equation takes place when the variable shows up in the exponential function. This can be a frightening topic for students, but with a bit of direction and practice, exponential equations can be worked out easily.
This article post will discuss the definition of exponential equations, kinds of exponential equations, steps to work out 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 include 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 primary things to bear in mind for when you seek to establish 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 first thing you must notice is that the variable, x, is in an exponent. Thereafter thing you should not 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 flipside, look at this equation:
y = 2x + 5
Once again, the primary thing you must observe is that the variable, x, is an exponent. The second thing you should note is that there are no other value that have the variable in them. This means that this equation IS exponential.
You will run into exponential equations when you try solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in math and perform a critical role in figuring out many math questions. Therefore, it is crucial to fully grasp what exponential equations are and how they can be used as you go ahead in mathematics.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three major kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with different bases on each sides, but they can be made similar using properties of the exponents. We will show some examples below, but by changing the bases the equal, you can observe the described steps as the first case.
3) Equations with different bases on each sides that is unable to be made the similar. These are the most difficult to work out, but it’s feasible utilizing the property of the product rule. By raising both factors to identical power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two new equations identical to one another and solve for the unknown variable. This article does not contain logarithm solutions, but we will tell you where to get assistance at the very last of this blog.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now understand how to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we need to ensue to work on exponential equations.
First, we must recognize the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them through standard algebraic rules.
Lastly, we have to work on the unknown variable. Once 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 check out a few examples to note how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are identical. Hence, all you are required to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
Now, we replace 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 problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. Despite that, both sides are powers of two. In essence, the working comprises of breaking down respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the ultimate answer:
28=22x-10
Carry out algebra to solve for x in the exponents as we performed in the last example.
8=2x-10
x=9
We can recheck our answer by altering 9 for x in the initial equation.
256=49−5=44
Keep searching for examples and questions over the internet, and if you utilize the rules of exponents, you will inturn master of these theorems, figuring out almost all exponential equations without issue.
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Working on problems with exponential equations can be tough in absence support. While this guide take you through the essentials, you still may find questions or word questions that make you stumble. Or possibly you require some extra assistance as logarithms come into play.
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