Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable shows up in the exponential function. This can be a frightening topic for kids, but with a some of instruction and practice, exponential equations can be determited easily.
This blog post will discuss the explanation of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with solutions. Let's get started!
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 look for when you seek to figure out 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 (besides the exponent)
For example, look at this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. Thereafter thing you must not is that there is another 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
One more time, the first thing you must note is that the variable, x, is an exponent. Thereafter thing you should note is that there are no other value that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when you try solving diverse calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are very important in mathematics and perform a critical duty in working out many computational problems. Thus, it is crucial to fully understand what exponential equations are and how they can be used as you go ahead in arithmetic.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable ordinary in everyday life. There are three main kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to solve, as we can easily set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same using rules of the exponents. We will take a look at some examples below, but by changing the bases the same, you can follow the described steps as the first case.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the most difficult to work out, but it’s feasible utilizing the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can set 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 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 ensuing these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to work on exponential equations.
Primarily, we must identify the base and exponent variables within the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Then, we can work on them utilizing standard algebraic techniques.
Third, we have to work on the unknown variable. Once we have solved for the variable, we can put this value back into our first equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to see how these steps work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are identical. Hence, all you are required to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we substitute 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 further complex sum. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. Despite that, both sides are powers of two. As such, the solution includes breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the ultimate answer:
28=22x-10
Carry out algebra to work out the x in the exponents as we did in the prior example.
8=2x-10
x=9
We can recheck our work by replacing 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and questions online, and if you utilize the rules of exponents, you will become a master of these concepts, working out almost all exponential equations without issue.
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Solving problems with exponential equations can be tricky without guidance. While this guide take you through the essentials, you still may encounter questions or word questions that make you stumble. Or possibly you require some further help as logarithms come into the scenario.
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