Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a scary topic for kids, but with a some of direction and practice, exponential equations can be determited simply.
This article post will talk about the definition of exponential equations, types of exponential equations, process to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The first step to solving an exponential equation is knowing when you have one.
Definition
Exponential equations are equations that have 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 bear in mind for when trying 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, 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 should observe is that there is additional 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
One more time, the first thing you must note is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more terms that includes any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when working on diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are very important in mathematics and play a pivotal role in figuring out many math questions. Thus, it is important to fully grasp what exponential equations are and how they can be utilized as you move 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 major types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations same as each other and work out for the unknown variable.
2) Equations with different bases on both sides, but they can be created similar using rules of the exponents. We will show some examples below, but by changing the bases the equal, you can observe the same steps as the first case.
3) Equations with variable bases on each sides that is impossible to be made the similar. These are the trickiest to figure out, but it’s attainable using the property of the product rule. By raising both factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can set the two latest equations identical to one another and solve for the unknown variable. This blog do not include logarithm solutions, but we will tell you where to get assistance at the very last of this blog.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now move on to how to work on any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
We have three steps that we are going 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 are in common base. Thereafter, we can work on them through standard algebraic techniques.
Third, we have to figure out the unknown variable. Once we have figured out the variable, we can plug this value back into our initial equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to see how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can notice that all the bases are identical. Therefore, all you are required to do is to restate the exponents and solve utilizing algebra:
y+1=3y
y=½
So, we substitute the value of y in the given equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex sum. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. By itself, the solution comprises of breaking down respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the final answer:
28=22x-10
Apply algebra to figure out x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can recheck our answer by replacing 9 for x in the first equation.
256=49−5=44
Keep looking for examples and problems on the internet, and if you use the laws of exponents, you will turn into a master of these concepts, solving most exponential equations without issue.
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