Exponential EquationsExplanation, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for children, but with a some of instruction and practice, exponential equations can be worked out easily.
This article post will discuss the explanation of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is knowing 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 key items to keep in mind for when trying to determine if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is no other term that has the variable in it (besides the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you should notice is that the variable, x, is in an exponent. The second thing you should notice is that there is another term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Yet again, the primary thing you should notice is that the variable, x, is an exponent. The second thing you must note is that there are no other value that have the variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in math and play a pivotal role in figuring out many computational problems. Hence, it is crucial to fully understand what exponential equations are and how they can be used as you move 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 major types of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can simply set the two equations equal to each other and figure out for the unknown variable.
2) Equations with different 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 changing the bases the equal, you can observe the described steps as the first event.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the toughest to work out, but it’s attainable using the property of the product rule. By increasing two or more factors to the same 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 each other and work on the unknown variable. This blog does not contain logarithm solutions, but we will let you know where to get assistance at the closing parts of this article.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now move on to how to solve any equation by following these easy procedures.
Steps for Solving Exponential Equations
We have three steps that we need to follow to work on exponential equations.
First, we must recognize the base and exponent variables in the equation.
Second, we need to rewrite an exponential equation, so all terms are in common base. Thereafter, we can work on them using standard algebraic methods.
Lastly, we have to figure out the unknown variable. Since we have solved for the variable, we can plug this value back into our first equation to discover 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 practice.
First, we will work on the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Therefore, all you are required to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
Now, 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 further complicated question. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. Despite that, both sides are powers of two. As such, the working includes breaking down respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the ultimate result:
28=22x-10
Apply algebra to figure out x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our answer by substituting 9 for x in the first equation.
256=49−5=44
Continue searching for examples and questions online, and if you use the properties of exponents, you will turn into a master of these theorems, solving most exponential equations without issue.
Better Your Algebra Skills with Grade Potential
Solving questions with exponential equations can be tricky without help. Even though this guide goes through the basics, you still may encounter questions or word questions that might stumble you. Or maybe you require some extra assistance as logarithms come into play.
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