Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation takes place when the variable shows up in the exponential function. This can be a scary topic for kids, but with a bit of direction and practice, exponential equations can be worked out easily.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, proceduce to figure out 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 understanding 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 major things to keep in mind for when trying to determine 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, take a look at this equation:
y = 3x2 + 7
The most important thing you should note is that the variable, x, is in an exponent. Thereafter thing you must notice is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
Yet again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you must note 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 working on different calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are crucial in arithmetic and play a pivotal responsibility in figuring out many computational problems. Therefore, it is crucial to completely grasp what exponential equations are and how they can be utilized 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 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 solve, as we can easily set the two equations equivalent as each other and work out 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 take a look at some examples below, but by converting the bases the same, you can follow the same steps as the first case.
3) Equations with variable bases on each sides that is impossible to be made the same. These are the toughest to solve, but it’s feasible utilizing 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 are done, we can determine the two latest equations identical to one another and work on the unknown variable. This blog does not contain logarithm solutions, but we will let you know where to get guidance at the very last of this blog.
How to Solve Exponential Equations
Knowing the explanation and kinds of exponential equations, we can now learn to solve any equation by following these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to work on exponential equations.
First, we must recognize the base and exponent variables within the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them using standard algebraic methods.
Lastly, we have to solve for the unknown variable. Once we have solved for the variable, we can put this value back into our original equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to see how these procedures work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are identical. Therefore, all you have to do is to rewrite the exponents and work on them using algebra:
y+1=3y
y=½
So, we change the value of y in the respective equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complicated question. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a identical base. However, both sides are powers of two. As such, the solution comprises of decomposing 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 find the final result:
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 recheck our work by replacing 9 for x in the original equation.
256=49−5=44
Continue searching for examples and problems on the internet, and if you use the laws of exponents, you will inturn master of these theorems, figuring out almost all exponential equations with no issue at all.
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Working on problems with exponential equations can be tricky in absence guidance. While this guide covers the essentials, you still might face questions or word questions that make you stumble. Or perhaps you desire some additional help as logarithms come into play.
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