Exponential EquationsExplanation, Solving, and Examples
In math, 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 instruction and practice, exponential equations can be solved easily.
This blog post will talk about the explanation of exponential equations, types of exponential equations, process to work out exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The first step to solving an exponential equation is understanding 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 bear 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 no other term that has the variable in it (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you should note is that the variable, x, is in an exponent. The second thing you should observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, check out 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 notice is that there are no more value that have the variable in them. This signifies that this equation IS exponential.
You will come upon exponential equations when working on different calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in arithmetic and play a critical role in working out many math problems. Hence, it is crucial 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 come in the exponent of an exponential equation. Exponential equations are remarkable common 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 most convenient to solve, as we can simply set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with different bases on each sides, but they can be created the same utilizing properties of the exponents. We will show some examples below, but by changing the bases the equal, you can follow the described steps as the first case.
3) Equations with different bases on each sides that cannot be made the similar. These are the trickiest to work out, but it’s possible using 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 are done, we can set the two new equations equal to each other and work on the unknown variable. This article do not contain logarithm solutions, but we will let you know where to get assistance at the end of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now move on to how to solve any equation by following these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to work on exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Second, we need to rewrite an exponential equation, so all terms are in common base. Then, we can work on them using standard algebraic methods.
Lastly, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our original equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to note how these procedures work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are identical. Therefore, all you are required 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 real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complex problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a identical base. But, both sides are powers of two. As such, the solution comprises of decomposing respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final result:
28=22x-10
Perform algebra to figure out x in the exponents as we did in the last example.
8=2x-10
x=9
We can recheck our answer by substituting 9 for x in the first equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you use the properties of exponents, you will inturn master of these concepts, working out almost all exponential equations without issue.
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