Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a scary topic for students, but with a some of direction and practice, exponential equations can be worked out quickly.
This blog post will discuss the explanation of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The primary step to work on an exponential equation is determining when you have one.
Definition
Exponential equations are equations that consist of 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 things 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 only one term that has the variable in it (aside from the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. The second thing you must 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 flipside, take a look at this equation:
y = 2x + 5
One more time, the first thing you should note is that the variable, x, is an exponent. Thereafter 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 across exponential equations when solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in math and play a critical duty in figuring out many computational questions. Therefore, it is important to completely understand what exponential equations are and how they can be used as you progress in arithmetic.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three primary types of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can easily set the two equations equivalent as each other and solve for the unknown variable.
2) Equations with different bases on each sides, but they can be made similar utilizing rules of the exponents. We will take a look at some examples below, but by converting the bases the same, you can observe the described steps as the first event.
3) Equations with distinct bases on both sides that is unable to be made the similar. These are the most difficult to work out, but it’s feasible through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can resolute the two latest equations identical to each other and figure out the unknown variable. This article does not include logarithm solutions, but we will let you know where to get help at the end of this article.
How to Solve Exponential Equations
From the explanation and kinds of exponential equations, we can now understand how to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to ensue to solve exponential equations.
First, we must recognize the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Then, we can work on them using standard algebraic rules.
Lastly, we have to solve for 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 take a loot at a few examples to note how these process work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can observe that both bases are identical. Thus, all you are required to do is to rewrite the exponents and solve through algebra:
y+1=3y
y=½
Right away, we substitute the value of y in the specified equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex problem. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. Despite that, both sides are powers of two. As such, the working comprises of breaking down respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to find the ultimate result:
28=22x-10
Carry out algebra to work out the x in the exponents as we did in the last example.
8=2x-10
x=9
We can verify our work by replacing 9 for x in the first equation.
256=49−5=44
Continue searching for examples and problems online, and if you utilize the laws of exponents, you will inturn master of these theorems, figuring out most exponential equations with no issue at all.
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Solving problems with exponential equations can be tough with lack of help. While this guide goes through the basics, you still might encounter questions or word problems that might stumble you. Or possibly you require some further assistance as logarithms come into play.
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