Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a scary topic for kids, but with a bit of direction and practice, exponential equations can be worked out easily.
This blog post will discuss the definition of exponential equations, kinds of exponential equations, process to solve exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining 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 primary 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 only one term that has the variable in it (in addition of 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 means that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the first thing you should note is that the variable, x, is an exponent. The second thing you must note is that there are no other value that have the variable in them. This implies that this equation IS exponential.
You will come across exponential equations when working on various calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are essential in math and perform a central role in figuring out many mathematical problems. Thus, it is important to fully understand what exponential equations are and how they can be utilized as you go ahead in mathematics.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three main types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with dissimilar 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 changing the bases the equal, you can follow the exact steps as the first instance.
3) Equations with variable bases on each sides that cannot be made the same. These are the most difficult to work out, but it’s possible using the property of the product rule. By raising both factors to the same power, we can multiply the factors on each side and raise them.
Once we are done, we can set the two latest equations equal to each other and figure out the unknown variable. This article do not include logarithm solutions, but we will let you know where to get help at the very last of this blog.
How to Solve Exponential Equations
From 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 ensue to work on exponential equations.
First, we must identify the base and exponent variables within the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them using standard algebraic methods.
Third, we have to work on the unknown variable. Once we have solved for the variable, we can plug this value back into our first equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Therefore, all you have to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we replace the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complicated problem. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. But, both sides are powers of two. In essence, the working comprises of breaking down respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final 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 double-check our answer 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 properties of exponents, you will become a master of these concepts, figuring out almost all exponential equations with no issue at all.
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Solving questions with exponential equations can be tricky with lack of guidance. Even though this guide take you through the essentials, you still might find questions or word problems that might stumble you. Or maybe you need some further assistance as logarithms come into play.
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