Exponential EquationsExplanation, Workings, and Examples
In math, an exponential equation takes place when the variable shows up in the exponential function. This can be a terrifying topic for students, but with a some of direction and practice, exponential equations can be determited quickly.
This blog post will talk about the definition of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's get started!
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 instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary items to keep in mind for when you seek 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, check out this equation:
y = 3x2 + 7
The first thing you should observe 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 means that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
One more time, the primary thing you should note is that the variable, x, is an exponent. The second thing you should observe is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when you try solving various calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are very important in arithmetic and play a central responsibility in solving many computational problems. Therefore, it is crucial to completely grasp what exponential equations are and how they can be utilized as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly common in everyday life. There are three main types of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to solve, as we can simply set the two equations same as each other and figure out for the unknown variable.
2) Equations with different bases on both sides, but they can be made the same employing properties of the exponents. We will put a few examples below, but by making the bases the same, you can follow the same steps as the first event.
3) Equations with distinct bases on both sides that is unable to be made the same. These are the most difficult to solve, but it’s attainable through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two latest equations equal to one another and figure out the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get help at the end of this article.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now understand how to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we need to ensue to solve exponential equations.
First, we must recognize the base and exponent variables in the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Then, we can work on them utilizing standard algebraic methods.
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 initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's check out some examples to see how these steps work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are the same. Therefore, all you need to do is to restate the exponents and figure them out through algebra:
y+1=3y
y=½
So, we change 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 observe this up with a further complex problem. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. However, both sides are powers of two. As such, the working includes decomposing 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 find the final answer:
28=22x-10
Carry out algebra to solve for x in the exponents as we did in the prior example.
8=2x-10
x=9
We can double-check our work by replacing 9 for x in the first equation.
256=49−5=44
Continue seeking for examples and questions on the internet, and if you utilize the rules of exponents, you will become a master of these theorems, working out almost all exponential equations without issue.
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Solving questions with exponential equations can be tricky with lack of guidance. While this guide take you through the fundamentals, you still might find questions or word problems that may hinder you. Or perhaps you desire some extra assistance as logarithms come into the scene.
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