Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation occurs 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 easily.
This article post will discuss the definition of exponential equations, kinds of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The primary step to work on an exponential equation is knowing when you have 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 most important thing you must note is that the variable, x, is in an exponent. The second thing you must observe is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the other hand, take a look at this equation:
y = 2x + 5
One more time, the primary thing you should observe is that the variable, x, is an exponent. The second thing you should observe is that there are no more value that consists of any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when working on different calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are very important in arithmetic and perform a critical duty in figuring out many computational questions. Therefore, it is important to completely grasp what exponential equations are and how they can be used as you progress in arithmetic.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly 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 work out for the unknown variable.
2) Equations with distinct bases on both sides, but they can be made the same using rules of the exponents. We will put a few examples below, but by changing the bases the equal, you can observe the exact steps as the first case.
3) Equations with distinct bases on both sides that cannot be made the same. These are the toughest to work out, but it’s possible through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can resolute the two latest equations identical to each other and work on the unknown variable. This blog do not contain logarithm solutions, but we will tell you where to get assistance at the very last of this article.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now move on to how to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to solve exponential equations.
Primarily, we must recognize the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them utilizing standard algebraic rules.
Third, we have to work on the unknown variable. Once we have solved for the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to observe how these steps 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. Thus, all you have to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Right away, we change the value of y in the respective equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex problem. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. However, both sides are powers of two. By itself, the working consists of breaking down respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the ultimate answer:
28=22x-10
Perform algebra to figure out x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can double-check our answer by substituting 9 for x in the initial equation.
256=49−5=44
Keep looking for examples and questions over the internet, and if you use the properties of exponents, you will inturn master of these concepts, figuring out almost all exponential equations with no issue at all.
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