Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation occurs when the variable shows up in the exponential function. This can be a terrifying topic for children, but with a bit of instruction and practice, exponential equations can be solved quickly.
This blog post will discuss the explanation of exponential equations, types of exponential equations, process to solve exponential equations, and examples with solutions. 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 key items to keep 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 (besides 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. The second thing you should notice is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
One more time, the first thing you must observe is that the variable, x, is an exponent. The second thing you must observe is that there are no other terms that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when you try solving different calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in arithmetic and perform a central responsibility in working out many math problems. Therefore, it is crucial to fully grasp what exponential equations are and how they can be utilized as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly ordinary in everyday life. There are three major kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can easily set the two equations equal to each other and figure out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made similar using properties of the exponents. We will show some examples below, but by making the bases the same, you can follow the same steps as the first event.
3) Equations with different bases on each sides that cannot be made the same. These are the toughest to solve, but it’s possible using the property of the product rule. By increasing both factors to identical power, we can multiply the factors on both side and raise them.
Once we are done, we can determine the two new equations identical to each other and solve for the unknown variable. This article do not cover logarithm solutions, but we will tell you where to get assistance at the closing parts of this article.
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 easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
First, we must determine the base and exponent variables in the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can solve them utilizing standard algebraic rules.
Third, we have to solve for the unknown variable. Since we have figured out the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to observe how these process work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that both bases are the same. Thus, all you need to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we change the value of y in the respective equation to corroborate 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 solution comprises of breaking down both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to find the ultimate answer:
28=22x-10
Carry out algebra to work out the x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our work by replacing 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems online, and if you utilize the properties of exponents, you will become a master of these theorems, figuring out most exponential equations without issue.
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