Exponential EquationsExplanation, Workings, and Examples
In math, an exponential equation arises when the variable appears in the exponential function. This can be a frightening topic for students, but with a some of instruction and practice, exponential equations can be solved easily.
This blog post will discuss the definition of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining 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 things to bear in mind for when trying to establish 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, look at this equation:
y = 3x2 + 7
The most important thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
One more time, the first thing you should observe is that the variable, x, is an exponent. Thereafter thing you must observe is that there are no more value that includes any variable in them. This means that this equation IS exponential.
You will run into exponential equations when solving different calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in mathematics and play a central role in figuring out many computational questions. Hence, it is crucial to completely grasp what exponential equations are and how they can be used as you go ahead in your math studies.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable ordinary in daily life. There are three main kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be made similar employing rules of the exponents. We will show some examples below, but by changing the bases the same, you can follow the same steps as the first case.
3) Equations with distinct bases on each sides that is unable to be made the same. These are the most difficult to solve, but it’s attainable through 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 have done this, we can determine the two latest equations identical to each other and work on the unknown variable. This article do not include logarithm solutions, but we will tell you where to get help 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 procedures.
Steps for Solving Exponential Equations
We have three steps that we are required to follow to work on exponential equations.
Primarily, we must determine the base and exponent variables inside the equation.
Next, we are required to rewrite an exponential equation, so all terms have a common base. Then, we can solve them utilizing standard algebraic rules.
Lastly, we have to work on the unknown variable. Once we have figured out the variable, we can plug this value back into our initial 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 procedures 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. Thus, all you are required to do is to restate the exponents and figure them out using algebra:
y+1=3y
y=½
Now, we change the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex 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. As such, the working consists of decomposing both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to come to the final result:
28=22x-10
Apply algebra to figure out x in the exponents as we did in the last example.
8=2x-10
x=9
We can recheck our answer by replacing 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and questions over the internet, and if you utilize the properties of exponents, you will turn into a master of these theorems, working out most exponential equations without issue.
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Solving problems with exponential equations can be tough in absence support. Even though this guide covers the basics, you still might face questions or word questions that make you stumble. Or possibly you require some additional guidance as logarithms come into the scenario.
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