Exponential EquationsExplanation, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a bit of direction and practice, exponential equations can be determited quickly.
This blog post will talk about the definition of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary step to solving an exponential equation is knowing when you are working with 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 attempting to determine 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 (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must notice is that the variable, x, is in an exponent. Thereafter thing you should not is that there is another term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Yet again, the primary thing you must note is that the variable, x, is an exponent. The second thing you must notice is that there are no more terms that have the variable in them. This means that this equation IS exponential.
You will come across exponential equations when solving different calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and perform a critical duty in figuring out many computational problems. Thus, it is critical to completely understand what exponential equations are and how they can be utilized as you go ahead in arithmetic.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly easy to find 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 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 rules of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the exact steps as the first event.
3) Equations with different bases on each sides that cannot be made the same. These are the toughest to solve, but it’s feasible through the property of the product rule. By raising both factors to the same power, we can multiply the factors on each side and raise them.
Once we have done this, we can determine the two new equations identical to one another and solve for the unknown variable. This blog do not cover logarithm solutions, but we will tell you where to get help at the closing parts of this blog.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now learn to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we need to ensue to work on exponential equations.
First, we must recognize the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Thereafter, we can solve them utilizing standard algebraic rules.
Lastly, we have to solve for the unknown variable. Since we have figured out the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to note how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Thus, all you need to do is to rewrite the exponents and solve through algebra:
y+1=3y
y=½
So, we substitute the value of y in the specified equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated question. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. But, both sides are powers of two. In essence, the working consists of decomposing respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to find the final result:
28=22x-10
Carry out algebra to figure out x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can verify our work by altering 9 for x in the original equation.
256=49−5=44
Continue searching for examples and questions over the internet, and if you utilize the rules of exponents, you will inturn master of these concepts, solving almost all exponential equations without issue.
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Solving questions with exponential equations can be difficult without support. Even though this guide take you through the fundamentals, you still may encounter questions or word problems that might stumble you. Or possibly you desire some additional help as logarithms come into play.
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