Exponential EquationsExplanation, Workings, 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 children, but with a bit of direction and practice, exponential equations can be worked out quickly.
This article post will talk about the definition of exponential equations, types of exponential equations, proceduce to work out 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 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 major things to bear in mind for when attempting 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 (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The first thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should notice is that there is another term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Once again, the primary thing you must note is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no more terms that includes any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving different calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are very important in math and perform a pivotal responsibility in working out many mathematical questions. Thus, it is important to fully understand what exponential equations are and how they can be used as you move ahead in your math studies.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly ordinary in daily 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 easiest to work out, as we can easily 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 made similar employing rules of the exponents. We will put a few examples below, but by converting the bases the equal, you can observe the described steps as the first instance.
3) Equations with distinct bases on each sides that is unable to be made the similar. These are the trickiest to solve, but it’s possible using the property of the product rule. By raising two or more factors to identical power, we can multiply the factors on both side and raise them.
Once we have done this, we can set the two latest equations identical to one another and work on the unknown variable. This blog does not cover logarithm solutions, but we will tell you where to get assistance at the very last of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now learn to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to work on 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. Thereafter, we can solve them utilizing standard algebraic methods.
Lastly, we have to work on the unknown variable. Since we have figured out the variable, we can put this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to see how these process work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Therefore, all you are required to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
So, we replace the value of y in the given equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. But, both sides are powers of two. In essence, the working comprises of decomposing both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to find the final result:
28=22x-10
Carry out algebra to figure out x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can double-check our answer by substituting 9 for x in the original equation.
256=49−5=44
Continue seeking for examples and questions online, and if you utilize the properties of exponents, you will inturn master of these concepts, solving almost all exponential equations with no issue at all.
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Solving questions with exponential equations can be difficult without support. Even though this guide take you through the fundamentals, you still may find questions or word questions that might stumble you. Or possibly you need some additional guidance as logarithms come into the scene.
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