Exponential EquationsDefinition, Workings, and Examples
In math, an exponential equation takes place when the variable appears in the exponential function. This can be a scary topic for students, but with a some of instruction and practice, exponential equations can be determited easily.
This blog post will discuss the definition of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is understanding 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 primary items to look for when attempting to determine if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is no other term that has the variable in it (aside from the exponent)
For example, look at this equation:
y = 3x2 + 7
The first thing you must observe is that the variable, x, is in an exponent. The second thing you should notice 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 contrary, take a look at this equation:
y = 2x + 5
Once again, the first thing you must observe is that the variable, x, is an exponent. Thereafter thing you must observe is that there are no more value that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when solving various calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are crucial in math and perform a pivotal duty in solving many mathematical questions. Therefore, it is crucial to completely grasp what exponential equations are and how they can be used as you progress in mathematics.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly common in everyday life. There are three primary kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the easiest 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 made similar utilizing rules of the exponents. We will take a look at some examples below, but by changing the bases the equal, you can follow the exact steps as the first event.
3) Equations with different bases on each sides that is unable to be made the similar. These are the trickiest to solve, but it’s feasible through the property of the product rule. By raising two or more factors to similar power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two latest equations identical to one another and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get guidance 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 easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to solve exponential equations.
First, we must identify the base and exponent variables within the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them using standard algebraic methods.
Lastly, we have to work on 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 take a loot at a few examples to see how these process work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can see that both bases are the same. Thus, all you need to do is to restate the exponents and figure them out 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 complex problem. 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 working includes breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to find the final result:
28=22x-10
Carry out algebra to work out the x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can recheck our work by substituting 9 for x in the first equation.
256=49−5=44
Continue searching for examples and questions over the internet, and if you use the properties of exponents, you will become a master of these theorems, figuring out most exponential equations without issue.
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Solving questions with exponential equations can be tricky in absence guidance. Even though this guide take you through the fundamentals, you still might face questions or word questions that make you stumble. Or possibly you require some extra guidance as logarithms come into play.
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