Exponential EquationsExplanation, Workings, and Examples
In math, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a bit of direction and practice, exponential equations can be worked out simply.
This blog post will discuss the explanation of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The initial step to work on an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that have 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 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 not is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the first thing you must notice is that the variable, x, is an exponent. Thereafter thing you must observe is that there are no more terms that have the variable in them. This implies that this equation IS exponential.
You will come across exponential equations when solving different calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are very important in arithmetic and play a pivotal duty in working out many math questions. Therefore, it is crucial to completely grasp what exponential equations are and how they can be utilized as you go ahead in your math studies.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable common in daily life. There are three major types of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient to work out, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made similar employing properties of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the same steps as the first instance.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the most difficult to solve, but it’s feasible using the property of the product rule. By raising both factors to identical power, we can multiply the factors on each side and raise them.
Once we have done this, we can set the two new equations equal to each other and figure out the unknown variable. This blog do not cover logarithm solutions, but we will let you know where to get assistance at the end of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now understand how to solve any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to work on 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 have a common base. Subsequently, we can solve them using standard algebraic techniques.
Third, we have to solve for the unknown variable. Once we have figured out the variable, we can put this value back into our initial equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's look at some examples to observe how these procedures work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that both bases are identical. Hence, all you need to do is to rewrite 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 corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated question. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. As such, the solution includes decomposing both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to come to the ultimate result:
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 recheck our work by substituting 9 for x in the original equation.
256=49−5=44
Keep looking for examples and problems online, and if you use the properties of exponents, you will become a master of these theorems, figuring out most exponential equations with no issue at all.
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Working on questions with exponential equations can be difficult with lack of guidance. Although this guide take you through the fundamentals, you still may face questions or word questions that might stumble you. Or possibly you need some extra assistance as logarithms come into the scenario.
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