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 instruction and practice, exponential equations can be determited 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 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 have 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 items 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, take a look at this equation:
y = 3x2 + 7
The primary thing you should observe is that the variable, x, is in an exponent. Thereafter thing you must observe is that there is another term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
One more time, the primary thing you should note is that the variable, x, is an exponent. Thereafter thing you should note is that there are no more terms that consists of any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when you try solving various calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are essential in arithmetic and play a critical responsibility in figuring out many computational questions. Thus, it is important to completely grasp what exponential equations are and how they can be utilized as you go ahead in arithmetic.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly common in everyday life. There are three main kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with different bases on both sides, but they can be created the same using properties of the exponents. We will show some examples below, but by changing the bases the equal, you can observe the described steps as the first case.
3) Equations with distinct bases on each sides that cannot be made the similar. These are the toughest to solve, but it’s possible through 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 latest equations identical to each other and solve for the unknown variable. This blog does not include logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
After going through the explanation and kinds of exponential equations, we can now learn to solve any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
There are three steps that we are going to follow to work on exponential equations.
Primarily, we must recognize the base and exponent variables inside the equation.
Second, we need 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. Once we have solved for the variable, we can put this value back into our original equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at a few examples to see how these process work in practice.
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 substitute 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 more complicated question. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a identical base. But, 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 figure out this expression to conclude the ultimate answer:
28=22x-10
Perform algebra to figure out x in the exponents as we performed in the previous 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
Keep looking for examples and questions over the internet, and if you utilize the laws of exponents, you will turn into a master of these concepts, figuring out almost all exponential equations without issue.
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