Exponential EquationsDefinition, Solving, and Examples
In mathematics, 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 instruction and practice, exponential equations can be worked out quickly.
This article post will talk about the definition of exponential equations, types of exponential equations, process to work out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that include 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 primary items to keep 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 only one term that has the variable in it (in addition of 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 notice is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the other hand, check out this equation:
y = 2x + 5
Once again, the primary thing you should note is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no more terms that consists of any variable in them. This implies that this equation IS exponential.
You will come across exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and perform a central role in solving many computational problems. Thus, 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 occur in the exponent of an exponential equation. Exponential equations are amazingly ordinary in everyday life. There are three main kinds of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the simplest to work out, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same using properties of the exponents. We will show some examples below, but by making the bases the same, you can follow the same steps as the first instance.
3) Equations with different bases on both sides that is unable to be made the similar. These are the toughest to solve, but it’s possible through the property of the product rule. By increasing both factors to the same power, we can multiply the factors on both side and raise them.
Once we have done this, we can set the two new equations equal to one another and work on the unknown variable. This article do not cover logarithm solutions, but we will tell you where to get guidance at the end of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now learn to work on any equation by following these simple steps.
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 in the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Thereafter, we can solve them using standard algebraic techniques.
Lastly, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our initial 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 note how these process work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that both bases are the same. Hence, all you need to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
Right away, we substitute the value of y in the respective equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex question. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. Despite that, 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 solve this expression to find the final result:
28=22x-10
Apply algebra to solve for x in the exponents as we performed in the last example.
8=2x-10
x=9
We can recheck our answer by substituting 9 for x in the initial equation.
256=49−5=44
Keep seeking for examples and questions on the internet, and if you utilize the properties of exponents, you will become a master of these theorems, solving almost all exponential equations without issue.
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