Exponential EquationsExplanation, Workings, and Examples
In arithmetic, an exponential equation takes place when the variable shows up in the exponential function. This can be a frightening topic for children, but with a some of direction and practice, exponential equations can be determited easily.
This blog post will discuss the definition of exponential equations, types of exponential equations, proceduce 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 are working with 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 major things to keep in mind for when attempting to determine 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, check out this equation:
y = 3x2 + 7
The most important thing you should notice is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the other hand, check out this equation:
y = 2x + 5
Once again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you must observe is that there are no more value that includes any variable in them. This means that this equation IS exponential.
You will come upon exponential equations when you try solving various calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are essential in mathematics and perform a critical role in solving many math questions. Hence, it is important to completely grasp what exponential equations are and how they can be used as you move ahead in your math studies.
Kinds of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable easy to find in everyday life. There are three main kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient to solve, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created similar utilizing properties of the exponents. We will take a look at some examples below, but by making the bases the same, you can observe the described steps as the first instance.
3) Equations with different bases on both sides that cannot be made the same. These are the trickiest to work out, but it’s attainable utilizing 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 one another and solve for the unknown variable. This article does not cover logarithm solutions, but we will let you know where to get guidance 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 move on to how to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we are going to ensue to solve exponential equations.
First, we must determine the base and exponent variables in the equation.
Next, we are required to rewrite an exponential equation, so all terms are in common base. Then, we can solve them using standard algebraic techniques.
Third, we have to figure out the unknown variable. Now that we have figured out the variable, we can plug this value back into our original equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to observe how these steps work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Hence, all you need to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
So, we change the value of y in the specified equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex problem. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. However, both sides are powers of two. By itself, the working includes breaking down both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the ultimate result:
28=22x-10
Apply 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
Keep searching for examples and problems on the internet, and if you use the rules of exponents, you will turn into a master of these concepts, solving almost all exponential equations with no issue at all.
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