Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation arises when the variable shows up in the exponential function. This can be a frightening topic for kids, but with a bit of direction and practice, exponential equations can be solved simply.
This blog post will discuss the explanation of exponential equations, kinds of exponential equations, proceduce to work out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The first step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that consist of 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 look 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 (aside from the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should observe 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
One more time, the primary thing you should notice is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other value that consists of any variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when you try solving various calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are very important in arithmetic and perform a pivotal role in working out many math problems. Thus, it is crucial to completely understand what exponential equations are and how they can be used as you move ahead in mathematics.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable easy to find in daily life. There are three primary kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can simply set the two equations equivalent as each other and figure out for the unknown variable.
2) Equations with different bases on both sides, but they can be created the same employing rules of the exponents. We will take a look at some examples below, but by changing the bases the equal, you can observe the described steps as the first event.
3) Equations with different bases on each sides that is impossible to be made the same. These are the trickiest 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 set the two new equations identical to one another and figure out 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
From the explanation and types of exponential equations, we can now learn to solve any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
Primarily, we must recognize the base and exponent variables in the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them utilizing standard algebraic methods.
Lastly, we have to figure out the unknown variable. Once we have figured out the variable, we can plug this value back into our original equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these procedures work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Hence, all you need to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
So, we replace the value of y in the given equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a identical base. However, both sides are powers of two. By itself, the solution comprises of 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 result:
28=22x-10
Carry out algebra to work out the x in the exponents as we did in the previous example.
8=2x-10
x=9
We can verify our answer by substituting 9 for x in the first equation.
256=49−5=44
Continue seeking for examples and problems online, and if you use the rules of exponents, you will become a master of these theorems, working out most exponential equations with no issue at all.
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