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 students, but with a some of instruction and practice, exponential equations can be solved easily.
This article post will discuss the definition of exponential equations, kinds of exponential equations, steps to work out exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The first step to figure out 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 you seek 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 no other term that has the variable in it (aside from the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you must notice is that the variable, x, is in an exponent. The second thing you should notice is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
Once again, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more value that have the variable in them. This means that this equation IS exponential.
You will come across exponential equations when you try solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in math and perform a critical duty in solving many mathematical questions. Thus, it is critical to completely grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly common in everyday life. There are three main kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient to solve, 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 utilizing rules of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the described steps as the first event.
3) Equations with distinct bases on both sides that cannot be made the same. These are the most difficult to work out, but it’s feasible using the property of the product rule. By raising both factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can determine the two new equations equal to each other and solve for the unknown variable. This article 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
Knowing the explanation and types of exponential equations, we can now learn to work on any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Then, we can work on them utilizing standard algebraic rules.
Lastly, we have to figure out the unknown variable. Now that we have figured out the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to observe how these procedures work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Therefore, all you are required to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
So, we replace 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 observe this up with a further complex sum. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. But, both sides are powers of two. By itself, the solution comprises of breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to find the ultimate result:
28=22x-10
Perform algebra to work out the x in the exponents as we did in the prior example.
8=2x-10
x=9
We can recheck our work by altering 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and problems on the internet, and if you use the rules of exponents, you will turn into a master of these concepts, solving most exponential equations without issue.
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Working on questions with exponential equations can be tough without guidance. Even though this guide covers the basics, you still may encounter questions or word questions that make you stumble. Or possibly you desire some extra help as logarithms come into play.
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