Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a frightening topic for kids, but with a bit of instruction and practice, exponential equations can be determited quickly.
This blog post will discuss the definition of exponential equations, types of exponential equations, process to work 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 understanding 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 key things to keep in mind 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, take a look at this equation:
y = 3x2 + 7
The first thing you should note is that the variable, x, is in an exponent. The second thing you should observe is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
One more time, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you must notice is that there are no more value that consists of any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving various calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in arithmetic and play a critical role in figuring out many math questions. Therefore, it is important to fully understand what exponential equations are and how they can be utilized as you progress in arithmetic.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three major kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the most convenient to work out, as we can easily set the two equations equivalent as each other and solve for the unknown variable.
2) Equations with different bases on both sides, but they can be made the same employing properties of the exponents. We will show some examples below, but by making the bases the equal, you can observe the exact steps as the first instance.
3) Equations with different bases on each sides that is unable to be made the same. These are the trickiest to work out, but it’s feasible utilizing the property of the product rule. By increasing both factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two latest equations equal to one another and work on the unknown variable. This blog does not cover 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 explanation and types of exponential equations, we can now understand how to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to solve 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. Then, we can work on them using standard algebraic rules.
Third, we have to work on 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 check out some examples to observe how these process work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can observe that all the bases are identical. Therefore, all you have to do is to restate the exponents and figure them out through algebra:
y+1=3y
y=½
Now, we substitute the value of y in the given 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 sum. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. Despite that, both sides are powers of two. As such, the working includes decomposing respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the ultimate answer:
28=22x-10
Perform algebra to work out the x in the exponents as we did in the previous example.
8=2x-10
x=9
We can verify our workings by altering 9 for x in the first equation.
256=49−5=44
Continue searching for examples and questions on the internet, and if you utilize the properties of exponents, you will turn into a master of these theorems, solving most exponential equations with no issue at all.
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Solving problems with exponential equations can be tricky without support. Even though this guide covers the fundamentals, you still may encounter questions or word problems that make you stumble. Or possibly you need some additional help as logarithms come into the scenario.
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