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 children, but with a some of instruction and practice, exponential equations can be solved easily.
This article post will talk about the explanation of exponential equations, types of exponential equations, process to figure out exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The first 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 example, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two key items 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 no other term that has the variable in it (besides the exponent)
For example, check out 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 not is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
One more time, 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 value that have the variable in them. This means 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 essential in mathematics and perform a pivotal role in solving many math questions. Therefore, it is important to completely grasp what exponential equations are and how they can be utilized as you go ahead in arithmetic.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly ordinary in daily life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the simplest to work out, as we can simply set the two equations same as each other and work out for the unknown variable.
2) Equations with distinct 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 follow the described steps as the first case.
3) Equations with variable bases on each sides that is impossible to be made the same. These are the trickiest to work out, but it’s feasible through the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two latest equations equal to each other and figure out the unknown variable. This article do not cover logarithm solutions, but we will tell you where to get assistance at the very last of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now move on to how to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we are required to follow to work on exponential equations.
First, we must determine the base and exponent variables in the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Then, we can solve them using standard algebraic methods.
Lastly, we have to solve for the unknown variable. Once we have figured out the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to see how these steps work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can notice that all the bases are the same. Thus, all you are required to do is to restate the exponents and solve utilizing algebra:
y+1=3y
y=½
Now, we change the value of y in the given equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex sum. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. However, both sides are powers of two. By itself, the working consists of breaking down respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final answer:
28=22x-10
Apply algebra to work out the x in the exponents as we did in the last example.
8=2x-10
x=9
We can verify our work by replacing 9 for x in the first equation.
256=49−5=44
Continue seeking for examples and problems online, and if you utilize the laws of exponents, you will turn into a master of these theorems, solving almost all exponential equations without issue.
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