Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a terrifying topic for kids, but with a some of direction and practice, exponential equations can be worked out simply.
This article post will discuss the definition of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is determining when you have one.
Definition
Exponential equations are equations that include 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 major things 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, look at this equation:
y = 3x2 + 7
The primary thing you should note is that the variable, x, is in an exponent. Thereafter thing you should not is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the flipside, check out this equation:
y = 2x + 5
Yet again, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other terms that have the variable in them. This implies that this equation IS exponential.
You will run into exponential equations when you try solving different calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in math and perform a pivotal role in working out many math questions. Thus, it is crucial to completely understand what exponential equations are and how they can be utilized as you go ahead in mathematics.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable easy to find in everyday life. There are three major kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations same as each other and figure out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made similar employing rules of the exponents. We will take a look at some examples below, but by making the bases the equal, you can observe the same steps as the first event.
3) Equations with different bases on both sides that cannot be made the same. These are the trickiest to solve, but it’s possible using 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 set the two latest equations identical to one another and solve for the unknown variable. This blog do not contain logarithm solutions, but we will tell you where to get help at the closing parts of this article.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now move on to how to solve any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to work on exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Then, we can work on them utilizing standard algebraic techniques.
Third, we have to work on the unknown variable. Now that we have figured out the variable, we can put this value back into our initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's look at some examples to see how these procedures work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Therefore, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
Right away, we substitute 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 question. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. Despite that, both sides are powers of two. By itself, the solution consists of decomposing respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the ultimate answer:
28=22x-10
Perform algebra to figure out 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
Keep looking for examples and questions over the internet, and if you utilize the laws of exponents, you will inturn master of these concepts, solving most exponential equations without issue.
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