Exponential EquationsDefinition, Workings, 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 bit of direction and practice, exponential equations can be solved simply.
This blog post will talk about the explanation of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is knowing when you have one.
Definition
Exponential equations are equations that have 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 trying to figure out if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is only one term that has the variable in it (besides the exponent)
For example, look at this equation:
y = 3x2 + 7
The primary thing you must note is that the variable, x, is in an exponent. Thereafter thing you should not is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Once again, the first thing you should note is that the variable, x, is an exponent. The second thing you must note is that there are no other value that includes any variable in them. This means that this equation IS exponential.
You will come across exponential equations when working on various calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are essential in arithmetic and play a central responsibility in figuring out many computational questions. Hence, it is critical to completely understand what exponential equations are and how they can be utilized as you progress in arithmetic.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly common in daily life. There are three primary kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created the same using rules of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the same steps as the first case.
3) Equations with different bases on each sides that is impossible to be made the similar. These are the trickiest to work out, but it’s attainable utilizing 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 set the two new equations equal to one another and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get guidance at the very last of this blog.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now learn to solve any equation by following these simple procedures.
Steps for Solving Exponential Equations
There are 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 need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them using standard algebraic rules.
Third, we have to figure out the unknown variable. Since we have figured out the variable, we can plug 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 some examples to observe how these procedures work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that both bases are identical. Thus, all you need to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
So, 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 follow 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 does not share a similar base. Despite that, both sides are powers of two. As such, the working includes decomposing both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the final answer:
28=22x-10
Perform algebra to figure out x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can double-check our workings by substituting 9 for x in the original equation.
256=49−5=44
Continue searching for examples and problems over the internet, and if you utilize the properties of exponents, you will turn into a master of these theorems, solving almost all exponential equations with no issue at all.
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Working on questions with exponential equations can be difficult in absence support. Even though this guide take you through the basics, you still may find questions or word questions that make you stumble. Or possibly you require some extra guidance as logarithms come into the scenario.
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