Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation takes place when the variable shows up in the exponential function. This can be a frightening topic for kids, but with a bit of direction and practice, exponential equations can be determited easily.
This blog post will talk about the explanation of exponential equations, types of exponential equations, steps to work out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary 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 items to look for when attempting 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 (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The first thing you must notice 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 signifies that this equation is NOT exponential.
On the flipside, 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. Thereafter thing you must observe is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when you try solving various calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in mathematics and play a critical role in figuring out many mathematical questions. Therefore, it is important to fully grasp what exponential equations are and how they can be used as you move ahead in your math studies.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in everyday life. There are three major kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can simply set the two equations equal to each other and figure out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be created similar using properties of the exponents. We will show some examples below, but by converting the bases the equal, you can observe the described steps as the first instance.
3) Equations with distinct bases on both sides that cannot be made the same. These are the trickiest to figure out, but it’s possible utilizing the property of the product rule. By increasing two or more factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can determine the two latest equations equal to one another and solve for the unknown variable. This article do not cover logarithm solutions, but we will let you know where to get assistance at the closing parts of this blog.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now move on to how to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
We have three steps that we need to follow to work on exponential equations.
First, we must determine the base and exponent variables inside the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Thereafter, we can work on them using standard algebraic methods.
Third, we have to figure out the unknown variable. Since we have figured out the variable, we can plug this value back into our first equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's check out some examples to observe how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that both bases are the same. Thus, all you have to do is to restate the exponents and figure them out utilizing algebra:
y+1=3y
y=½
Right away, we substitute the value of y in the specified equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complex sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. Despite that, both sides are powers of two. By itself, the solution comprises of breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to conclude the ultimate answer:
28=22x-10
Perform algebra to work out the x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can recheck our answer by replacing 9 for x in the initial equation.
256=49−5=44
Keep searching for examples and questions on the internet, and if you utilize the properties of exponents, you will become a master of these theorems, working out most exponential equations with no issue at all.
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