Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation arises when the variable appears in the exponential function. This can be a frightening topic for children, but with a bit of instruction and practice, exponential equations can be determited easily.
This blog post will discuss the definition of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The primary step to work on an exponential equation is knowing when you have 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 major things to look for when attempting to establish if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is no other term that has the variable in it (aside from the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you should 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 signifies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
One more time, the first thing you should notice is that the variable, x, is an exponent. The second thing you should notice is that there are no more terms that have the variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when you try solving diverse calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in math and perform a critical role in working out many computational questions. Therefore, it is critical to fully understand what exponential equations are and how they can be used as you progress in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable ordinary in everyday life. There are three major kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can easily set the two equations equivalent as each other and figure out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be made similar using rules of the exponents. We will take a look at some examples below, but by converting the bases the same, you can observe the same steps as the first instance.
3) Equations with distinct bases on each sides that is impossible to be made the same. These are the trickiest to work out, 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 each side and raise them.
Once we have done this, we can set the two new equations identical to one another and work on the unknown variable. This blog 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 kinds of exponential equations, we can now learn to solve any equation by following these easy procedures.
Steps for Solving Exponential Equations
There are three steps that we are going to follow to work on exponential equations.
First, we must determine the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Then, we can work on them utilizing standard algebraic methods.
Lastly, we have to figure out the unknown variable. Since we have solved for the variable, we can plug this value back into our initial equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to note how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are identical. Thus, all you are required to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Right away, we change the value of y in the respective equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex question. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. In essence, the working comprises of breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to find the ultimate answer:
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 answer by altering 9 for x in the first equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you utilize the rules of exponents, you will inturn master of these theorems, working out almost all exponential equations with no issue at all.
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