Exponential EquationsExplanation, Solving, and Examples
In arithmetic, 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 talk about the explanation of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The first step to solving an exponential equation is knowing when you are working with one.
Definition
Exponential equations are equations that include the variable in an exponent. For example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary things to bear in mind for when trying to establish 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 observe is that the variable, x, is in an exponent. The second thing you must observe 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 other hand, take a look at this equation:
y = 2x + 5
One more time, the primary thing you must note is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no more terms that have the variable in them. This signifies that this equation IS exponential.
You will run into exponential equations when solving different calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are crucial in math and perform a pivotal duty in working out many computational problems. Hence, it is crucial to fully grasp what exponential equations are and how they can be used as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in daily life. There are three primary 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 easily set the two equations same as each other and work out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created similar employing properties of the exponents. We will put a few examples below, but by changing the bases the equal, you can follow the exact steps as the first case.
3) Equations with distinct bases on both sides that cannot be made the same. These are the toughest to solve, but it’s possible using the property of the product rule. By raising 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 equal to each other and work on the unknown variable. This blog does not contain logarithm solutions, but we will let you know where to get help at the end of this blog.
How to Solve Exponential Equations
After going through the explanation and kinds of exponential equations, we can now learn to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we are going to ensue to solve exponential equations.
Primarily, we must determine 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 solve them using standard algebraic techniques.
Third, we have to figure out the unknown variable. Now that we have solved for the variable, we can put this value back into our first equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to see how these steps work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are the same. Thus, all you need to do is to rewrite the exponents and work on them using algebra:
y+1=3y
y=½
Now, we replace the value of y in the specified 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 complicated problem. 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. By itself, the working consists of decomposing both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we work on this expression to come to the ultimate result:
28=22x-10
Apply algebra to solve for x in the exponents as we did in the prior example.
8=2x-10
x=9
We can double-check our workings by altering 9 for x in the initial equation.
256=49−5=44
Continue looking for examples and problems online, and if you utilize the rules of exponents, you will turn into a master of these concepts, working out most exponential equations without issue.
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