Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation arises when the variable shows up in the exponential function. This can be a scary topic for children, but with a some of direction and practice, exponential equations can be worked out easily.
This blog post will discuss the definition of exponential equations, types of exponential equations, steps to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The first step to figure out an exponential equation is knowing when you are working with 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 major items to bear in mind for when you seek to figure out 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 (aside from the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you must note is that the variable, x, is in an exponent. The second thing you must not is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the flipside, check out this equation:
y = 2x + 5
Once again, the primary thing you must note is that the variable, x, is an exponent. The second thing you must observe is that there are no other 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 mathematics and play a central duty in working out many math questions. Therefore, it is important to fully grasp what exponential equations are and how they can be utilized as you progress in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three major types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the most convenient to work out, as we can simply set the two equations same as each other and work out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created similar using properties of the exponents. We will show some examples below, but by converting the bases the same, you can observe the same steps as the first case.
3) Equations with different bases on both sides that is impossible to be made the same. These are the trickiest to work out, but it’s possible using the property of the product rule. By increasing both factors to similar power, we can multiply the factors on both side and raise them.
Once we are done, we can set the two latest equations identical to one another and solve for the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get guidance at the very last of this blog.
How to Solve Exponential Equations
Knowing the explanation and kinds of exponential equations, we can now learn to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
There are three steps that we are going to follow to work on exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Next, we are required to rewrite an exponential equation, so all terms have a common base. Then, we can solve them through standard algebraic rules.
Third, we have to figure out the unknown variable. Since we have solved for 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 look at a few examples to note how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are identical. Therefore, all you have to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we replace the value of y in the respective equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. But, both sides are powers of two. In essence, the working 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 solve for x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our work by altering 9 for x in the original equation.
256=49−5=44
Continue looking for examples and questions online, and if you use the properties of exponents, you will inturn master of these concepts, working out almost all exponential equations with no issue at all.
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Solving questions with exponential equations can be difficult with lack of help. Although this guide take you through the basics, you still might find questions or word problems that might stumble you. Or perhaps you require some extra help as logarithms come into play.
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