Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation arises when the variable shows up in the exponential function. This can be a terrifying topic for children, but with a bit of instruction and practice, exponential equations can be determited simply.
This blog post will talk about the definition of exponential equations, types of exponential equations, steps to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The initial step to solving an exponential equation is knowing 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 major items to keep in mind 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 (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you must observe 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 – not only in an exponent. This signifies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the primary thing you should note is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more terms that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when solving different calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are crucial in arithmetic and perform a pivotal duty in solving many math questions. Therefore, it is crucial to completely understand what exponential equations are and how they can be utilized as you move ahead in mathematics.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable common in daily life. There are three major kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the most convenient to solve, as we can easily set the two equations same as each other and figure out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made the same utilizing properties of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the described steps as the first case.
3) Equations with different bases on both sides that is unable to be made the similar. These are the trickiest to work out, but it’s feasible utilizing the property of the product rule. By increasing both factors to similar 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 one another and solve for the unknown variable. This blog do not cover logarithm solutions, but we will tell you where to get assistance at the closing parts of this article.
How to Solve Exponential Equations
After going through the explanation and types of exponential equations, we can now learn to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to work on exponential equations.
First, we must determine the base and exponent variables within the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Thereafter, we can solve them using standard algebraic rules.
Lastly, we have to solve for the unknown variable. Since we have solved for the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to note how these steps work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are the same. Hence, all you are required to do is to restate the exponents and work on them through algebra:
y+1=3y
y=½
Right away, we change the value of y in the respective equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complicated sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a identical base. However, both sides are powers of two. As such, the solution consists of breaking down respectively the 4 and the 256, and we can replace 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 work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our workings by altering 9 for x in the first equation.
256=49−5=44
Keep searching for examples and questions over the internet, and if you use the rules of exponents, you will inturn master of these theorems, solving almost all exponential equations without issue.
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Working on problems with exponential equations can be tough with lack of guidance. Even though this guide take you through the fundamentals, you still might find questions or word questions that may hinder you. Or perhaps you require some additional help as logarithms come into play.
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