Exponential EquationsDefinition, Workings, 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 bit of instruction and practice, exponential equations can be solved simply.
This blog post will talk about the definition of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The initial step to solving an exponential equation is determining 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 key things to keep in mind for when you seek 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 most important thing you should note is that the variable, x, is in an exponent. Thereafter thing you must not is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Once again, the first thing you must notice is that the variable, x, is an exponent. Thereafter thing you should note is that there are no other value that includes any variable in them. This means that this equation IS exponential.
You will run into exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are very important in arithmetic and play a central duty in working out many computational questions. Hence, it is important to fully understand what exponential equations are and how they can be utilized as you go ahead in your math studies.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable ordinary in everyday life. There are three primary types of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can easily set the two equations equivalent as each other and solve 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 take a look at some examples below, but by converting the bases the same, you can follow the exact steps as the first instance.
3) Equations with different bases on each sides that is impossible to be made the similar. These are the toughest to solve, but it’s possible using the property of the product rule. By increasing two or more factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two latest equations identical to one another and solve for the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
After going through the definition and types of exponential equations, we can now learn to solve any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to work on exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them through standard algebraic rules.
Third, we have to solve for the unknown variable. Now that we have solved for the variable, we can plug this value back into our first equation to find 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 procedures work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are the same. Therefore, all you have to do is to restate the exponents and work on them using algebra:
y+1=3y
y=½
So, 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 follow this up with a more complex sum. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. However, both sides are powers of two. In essence, the solution includes decomposing respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final answer:
28=22x-10
Apply algebra to work out the x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can double-check our work by altering 9 for x in the initial equation.
256=49−5=44
Keep looking for examples and problems on the internet, and if you use the rules of exponents, you will become a master of these theorems, figuring out most exponential equations without issue.
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