Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation occurs when the variable shows up in the exponential function. This can be a scary topic for students, but with a bit of direction and practice, exponential equations can be determited simply.
This article post will discuss the definition of exponential equations, types of exponential equations, steps to work out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The initial step to work on an exponential equation is determining when you are working with 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 primary things to bear in mind 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 no other term that has the variable in it (besides the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Once again, the first thing you should observe is that the variable, x, is an exponent. The second thing you should notice is that there are no more value that have the variable in them. This implies that this equation IS exponential.
You will come across exponential equations when solving different calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are crucial in mathematics and play a central duty in working out many math problems. Hence, it is important to completely grasp 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 amazingly ordinary in everyday life. There are three main types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the simplest to work out, as we can easily set the two equations same as each other and work out for the unknown variable.
2) Equations with distinct bases on both sides, but they can be made the same utilizing properties of the exponents. We will take a look at some examples below, but by changing the bases the same, you can observe the described steps as the first instance.
3) Equations with different bases on each sides that cannot be made the similar. These are the toughest to work out, but it’s possible through the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on each side and raise them.
Once we have done this, we can determine the two latest equations equal to one another and solve for the unknown variable. This blog do not contain logarithm solutions, but we will let you know where to get guidance at the closing parts of this blog.
How to Solve Exponential Equations
From the explanation and kinds of exponential equations, we can now understand how to work on any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to work on exponential equations.
First, we must identify the base and exponent variables inside the equation.
Second, we need to rewrite an exponential equation, so all terms are in common base. Thereafter, we can solve them through standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Since we have figured out the variable, we can put 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 a few examples to see how these steps work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that both bases are identical. Hence, all you have to do is to rewrite the exponents and work on them through algebra:
y+1=3y
y=½
So, we change the value of y in the specified equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. Despite that, both sides are powers of two. As such, the working includes breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the final answer:
28=22x-10
Perform algebra to work out the x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can verify our work by altering 9 for x in the original equation.
256=49−5=44
Keep searching for examples and problems on the internet, and if you use the rules of exponents, you will inturn master of these concepts, solving almost all exponential equations without issue.
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