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 direction and practice, exponential equations can be worked out simply.
This blog post will discuss the explanation of exponential equations, types of exponential equations, steps to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The initial step to work on an exponential equation is determining 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 major things to keep in mind for when attempting to figure out 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, take a look at this equation:
y = 3x2 + 7
The primary thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is another term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Yet again, the primary thing you should note is that the variable, x, is an exponent. The second thing you should note is that there are no more terms that includes any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when working on diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in math and play a central duty in solving many computational problems. Therefore, it is important to fully grasp what exponential equations are and how they can be utilized as you move ahead in your math studies.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three main kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with distinct bases on both sides, but they can be made similar utilizing properties of the exponents. We will put a few examples below, but by making the bases the same, you can follow the exact steps as the first case.
3) Equations with different bases on each sides that cannot be made the same. These are the most difficult to figure out, but it’s possible through the property of the product rule. By raising both factors to similar power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two latest equations equal to one another and solve for the unknown variable. This blog do not include logarithm solutions, but we will tell you where to get guidance at the closing parts of this article.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now move on to how to solve any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to solve 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. Then, we can work on them using standard algebraic methods.
Lastly, we have to solve for the unknown variable. Once we have figured out the variable, we can put this value back into our original equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's look at some 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. Hence, all you are required to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
So, we replace the value of y in the specified equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complicated problem. Let's work on 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. 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 figure out this expression to find the ultimate 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 verify our answer by substituting 9 for x in the original equation.
256=49−5=44
Keep looking for examples and questions on the internet, and if you use the rules of exponents, you will turn into a master of these concepts, solving most exponential equations with no issue at all.
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