Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a terrifying topic for children, but with a bit of direction and practice, exponential equations can be solved simply.
This article post will talk about the definition of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The initial step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that consist of the variable in an exponent. For example, 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 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 most important thing you must observe is that the variable, x, is in an exponent. The second thing you should observe is that there is another term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the other hand, take a look at this equation:
y = 2x + 5
Yet again, the first thing you should observe is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are very important in mathematics and play a critical role in solving many math questions. Therefore, it is critical to fully understand what exponential equations are and how they can be used as you go ahead in mathematics.
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 the same bases on both sides. This is the easiest 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 employing properties of the exponents. We will take a look at some examples below, but by making the bases the equal, you can observe the exact steps as the first event.
3) Equations with different bases on both sides that is unable to be made the similar. These are the toughest to work out, but it’s feasible 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 determine the two latest equations identical to each other and figure out the unknown variable. This article do not cover 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 types of exponential equations, we can now move on to how to work on any equation by following these easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to solve exponential equations.
First, we must identify the base and exponent variables inside the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Then, we can solve them through standard algebraic methods.
Lastly, we have to solve for the unknown variable. Since we have solved for the variable, we can put this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's look at a few examples to note how these steps work in practicality.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are the same. Thus, all you need to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
Right away, we substitute 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 complicated sum. 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. By itself, the working includes decomposing both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the final answer:
28=22x-10
Perform algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can double-check our workings by substituting 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems on the internet, and if you utilize the properties of exponents, you will inturn master of these concepts, figuring out almost all exponential equations without issue.
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Solving problems with exponential equations can be tough without help. Although this guide covers the fundamentals, you still may encounter questions or word problems that may hinder you. Or perhaps you need some further assistance as logarithms come into play.
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