Exponential EquationsDefinition, Solving, and Examples
In arithmetic, an exponential equation takes place when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a bit of instruction and practice, exponential equations can be solved easily.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The first step to figure out an exponential equation is knowing when you are working with 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 bear in mind 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, look at this equation:
y = 3x2 + 7
The most important thing you should observe is that the variable, x, is in an exponent. The second thing you must observe is that there is additional 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
One more time, the first thing you should observe is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more value that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when solving various calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are crucial in mathematics and play a pivotal responsibility in working out many math questions. Hence, it is important to fully understand what exponential equations are and how they can be used as you move ahead in your math studies.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three primary kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can easily set the two equations same as each other and solve 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 equal, you can follow the same steps as the first event.
3) Equations with different bases on each sides that is unable to be made the same. These are the toughest to figure out, but it’s attainable using the property of the product rule. By raising two or more factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two latest equations equal to one another and solve for the unknown variable. This article does not cover logarithm solutions, but we will tell you where to get guidance at the very last of this article.
How to Solve Exponential Equations
After going through the explanation and types of exponential equations, we can now learn to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
There are three steps that we are going to follow to solve exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them utilizing standard algebraic methods.
Third, we have to figure out the unknown variable. Once we have figured out the variable, we can plug this value back into our first equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's check out a few examples to see how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are identical. Hence, all you have to do is to restate the exponents and figure them out utilizing algebra:
y+1=3y
y=½
So, we substitute the value of y in the specified equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a further complicated question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. But, both sides are powers of two. As such, the solution comprises of breaking down respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the final result:
28=22x-10
Perform algebra to solve for x in the exponents as we did in the last 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 looking for examples and problems over the internet, and if you utilize the laws of exponents, you will turn into a master of these theorems, working out most exponential equations without issue.
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