Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for kids, but with a some of direction and practice, exponential equations can be worked out simply.
This article post will discuss the definition of exponential equations, kinds of exponential equations, proceduce to solve exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to work on an exponential equation is determining when you have one.
Definition
Exponential equations are equations that include 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 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 (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 notice is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Once again, the primary thing you should observe is that the variable, x, is an exponent. Thereafter thing you should note is that there are no more terms that consists of any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when working on different calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and perform a critical responsibility in working out many mathematical problems. Thus, it is important to fully understand what exponential equations are and how they can be utilized as you progress in your math studies.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly ordinary in daily life. There are three main 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 solve for the unknown variable.
2) Equations with different bases on each sides, but they can be made similar utilizing rules of the exponents. We will put a few examples below, but by converting the bases the equal, you can follow the same steps as the first case.
3) Equations with different bases on each sides that is unable to be made the similar. These are the most difficult to solve, but it’s attainable through the property of the product rule. By raising both factors to identical power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two new equations equal to each other and work on the unknown variable. This blog do not cover logarithm solutions, but we will let you know where to get help at the very last of this blog.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now learn to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to solve exponential equations.
Primarily, we must determine the base and exponent variables inside the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them utilizing standard algebraic rules.
Third, we have to solve for the unknown variable. Now that we have solved for the variable, we can plug this value back into our original equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's check out a few examples to observe how these procedures work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are identical. Hence, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
Right away, we change 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 more complex problem. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. As such, the working comprises of breaking down respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we figure out this expression to conclude the final answer:
28=22x-10
Apply algebra to solve for x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our workings by substituting 9 for x in the first equation.
256=49−5=44
Continue seeking for examples and questions on the internet, and if you use the rules of exponents, you will become a master of these theorems, working out most exponential equations with no issue at all.
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