Exponential EquationsDefinition, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable shows up in the exponential function. This can be a scary topic for children, but with a bit of instruction and practice, exponential equations can be determited easily.
This article post will talk about the definition of exponential equations, types of exponential equations, process to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to figure out an exponential equation is determining 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 primary items to look for when you seek to determine 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 (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you must notice is that the variable, x, is in an exponent. Thereafter thing you must observe 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, check out this equation:
y = 2x + 5
One more time, the first thing you must note is that the variable, x, is an exponent. Thereafter thing you must observe 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 you try solving 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 crucial to completely grasp what exponential equations are and how they can be used as you go ahead in mathematics.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly common in daily life. There are three primary kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the simplest to work out, as we can simply set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on both sides, but they can be made the same using properties of the exponents. We will show some examples below, but by changing the bases the same, you can observe the same steps as the first instance.
3) Equations with variable bases on each sides that is impossible to be made the same. These are the most difficult to figure out, but it’s possible through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two new equations equal to one another and work on the unknown variable. This blog do not include logarithm solutions, but we will tell you where to get assistance at the very last of this blog.
How to Solve Exponential Equations
After going through the definition and types of exponential equations, we can now learn to solve any equation by following these easy steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to work on exponential equations.
Primarily, we must determine the base and exponent variables inside the equation.
Next, we are required 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 solve for the unknown variable. Since we have solved for the variable, we can plug this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to see how these steps work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are the same. Hence, all you have to do is to restate the exponents and work on them through algebra:
y+1=3y
y=½
So, we change the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complex sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. However, both sides are powers of two. By itself, the solution 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 ultimate result:
28=22x-10
Perform algebra to work out the x in the exponents as we did in the prior example.
8=2x-10
x=9
We can recheck our work by substituting 9 for x in the initial equation.
256=49−5=44
Keep looking for examples and questions online, and if you utilize the properties of exponents, you will turn into a master of these concepts, working out almost all exponential equations without issue.
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