Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation takes place when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a some of instruction and practice, exponential equations can be worked out easily.
This blog post will talk about the definition of exponential equations, types of exponential equations, process to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The first step to solving an exponential equation is knowing when you have 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 key things to look for when you seek to establish 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, check out this equation:
y = 3x2 + 7
The most important thing you must note is that the variable, x, is in an exponent. Thereafter thing you should observe 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, take a look at this equation:
y = 2x + 5
Yet again, the first thing you must observe is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no more value that have the variable in them. This implies that this equation IS exponential.
You will run into exponential equations when working on different calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in mathematics and perform a critical duty in figuring out many mathematical problems. Hence, it is important to fully grasp what exponential equations are and how they can be utilized as you move ahead in your math studies.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly ordinary in daily life. There are three main types of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can easily set the two equations equal to each other and solve 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 show some examples below, but by making the bases the same, you can follow the same steps as the first event.
3) Equations with variable bases on each sides that cannot be made the same. These are the toughest to work out, but it’s possible using the property of the product rule. By raising two or more factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can determine the two new equations identical to each other and figure out the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get assistance at the closing parts of this blog.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now learn to solve any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to solve exponential equations.
First, we must recognize the base and exponent variables inside the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Thereafter, we can work on them using standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Once we have figured out the variable, we can plug this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at some examples to observe how these steps work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that both bases are the same. Therefore, all you need to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
Now, we replace the value of y in the specified equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complex problem. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a identical base. However, both sides are powers of two. As such, the solution consists of decomposing respectively 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 final result:
28=22x-10
Carry out algebra to work out the x in the exponents as we performed in the last example.
8=2x-10
x=9
We can verify our workings by substituting 9 for x in the initial equation.
256=49−5=44
Continue looking for examples and questions on the internet, and if you utilize the rules of exponents, you will inturn master of these theorems, solving most exponential equations with no issue at all.
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