Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for students, but with a bit of direction and practice, exponential equations can be determited quickly.
This article post will discuss the explanation of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that consist of the variable in an exponent. For example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two key things to bear in mind for when trying 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 only one 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 must observe is that the variable, x, is in an exponent. The second thing you must observe is that there is additional 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, take a look at this equation:
y = 2x + 5
Yet again, the first thing you should note is that the variable, x, is an exponent. Thereafter thing you must note is that there are no more terms that includes any variable in them. This means that this equation IS exponential.
You will run into exponential equations when solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are crucial in math and play a central duty in figuring out many mathematical questions. Hence, it is important to fully grasp what exponential equations are and how they can be utilized as you move ahead in arithmetic.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable ordinary in daily life. There are three main kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can easily set the two equations equivalent as each other and figure out for the unknown variable.
2) Equations with different bases on each sides, but they can be made the same employing properties of the exponents. We will show some examples below, but by making the bases the equal, you can follow the exact steps as the first case.
3) Equations with different bases on each sides that cannot be made the similar. These are the most difficult to work out, but it’s attainable utilizing the property of the product rule. By raising both factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can determine the two new 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 guidance at the end of this article.
How to Solve Exponential Equations
After going through the explanation and kinds of exponential equations, we can now learn to work on any equation by following these easy steps.
Steps for Solving Exponential Equations
There are three steps that we need to follow to solve exponential equations.
First, we must identify the base and exponent variables inside the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Then, we can solve them using standard algebraic methods.
Lastly, we have to solve for the unknown variable. Once we have figured out 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 look at a few examples to see how these procedures work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are identical. Hence, all you need to do is to rewrite the exponents and solve through algebra:
y+1=3y
y=½
Now, we change the value of y in the given equation to corroborate 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 can see, the sides of the equation does not share a similar base. But, both sides are powers of two. As such, the solution consists of breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to come to the ultimate result:
28=22x-10
Carry out algebra to work out the x in the exponents as we did in the last 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
Continue searching for examples and problems online, and if you utilize the laws of exponents, you will become a master of these theorems, figuring out most exponential equations without issue.
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