Exponential EquationsExplanation, Workings, and Examples
In arithmetic, an exponential equation takes place 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 determited simply.
This article post will discuss the definition of exponential equations, types of exponential equations, process to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The first step to figure out an exponential equation is determining when you have one.
Definition
Exponential equations are equations that have 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 items to look for when you seek to determine 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 (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The first thing you should note is that the variable, x, is in an exponent. Thereafter thing you should notice is that there is another term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you must note is that there are no other terms that consists of any variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when you try solving diverse calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are very important in arithmetic and play a central role in solving many mathematical problems. Hence, it is critical to fully grasp what exponential equations are and how they can be used as you progress in arithmetic.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable common in everyday life. There are three main kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the most convenient 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 created similar using rules of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the exact steps as the first event.
3) Equations with different bases on each sides that cannot be made the same. These are the trickiest to figure out, but it’s possible using the property of the product rule. By raising both factors to identical power, we can multiply the factors on both 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 article does not cover logarithm solutions, but we will let you know where to get help at the very last of this blog.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now understand how to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to work on exponential equations.
Primarily, we must recognize the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Thereafter, we can solve them utilizing standard algebraic techniques.
Third, we have to solve for the unknown variable. Now that 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 check out a few examples to note how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that both bases are identical. Hence, all you have to do is to restate the exponents and figure them out through algebra:
y+1=3y
y=½
Right away, we replace 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 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 do not share a common base. However, both sides are powers of two. In essence, the solution includes decomposing both 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 ultimate 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 double-check 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 utilize the rules of exponents, you will become a master of these concepts, solving almost all exponential equations without issue.
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