Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for students, but with a bit of direction and practice, exponential equations can be solved simply.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, process to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The primary step to solving an exponential equation is knowing when you have 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 key items to keep in mind for when trying 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 only one term that has the variable in it (in addition of the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you should note is that the variable, x, is in an exponent. Thereafter thing you must observe 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 contrary, look at this equation:
y = 2x + 5
One more time, the first thing you should notice is that the variable, x, is an exponent. The second thing you must notice is that there are no other terms that consists of any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when solving various calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are crucial in math and perform a critical responsibility in figuring out many computational problems. Therefore, it is crucial to fully understand what exponential equations are and how they can be utilized as you go ahead in arithmetic.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in everyday life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the easiest to solve, as we can simply set the two equations equal to each other and work out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be made the same using properties of the exponents. We will take a look at some examples below, but by converting the bases the equal, you can observe the described steps as the first event.
3) Equations with variable bases on each sides that is unable to be made the similar. These are the trickiest to figure out, but it’s attainable utilizing the property of the product rule. By raising both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can determine the two new equations equal to each other and solve for the unknown variable. This blog do not contain logarithm solutions, but we will let you know where to get guidance at the end 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 easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to solve exponential equations.
First, we must determine the base and exponent variables within the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them utilizing standard algebraic methods.
Third, we have to work on the unknown variable. Now that we have solved for the variable, we can plug 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 take a loot at some examples to see how these steps work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are identical. Therefore, all you are required to do is to restate the exponents and work on them using algebra:
y+1=3y
y=½
Right away, we change the value of y in the given 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 problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. However, both sides are powers of two. In essence, the solution includes breaking down respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to come to the final answer:
28=22x-10
Perform algebra to figure out x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can verify our workings by replacing 9 for x in the original equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you use the rules of exponents, you will inturn master of these theorems, working 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 help. Even though this guide take you through the fundamentals, you still may encounter questions or word problems that make you stumble. Or perhaps you desire some extra guidance as logarithms come into play.
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