Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation takes place when the variable appears in the exponential function. This can be a terrifying topic for children, but with a some of direction and practice, exponential equations can be solved quickly.
This article post will talk about the definition of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is understanding when you are working with one.
Definition
Exponential equations are equations that consist of 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 things to bear in mind for when attempting 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 no other term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you must note is that the variable, x, is in an exponent. Thereafter thing you should not is that there is another term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the other hand, take a look at this equation:
y = 2x + 5
Once again, the first thing you should notice is that the variable, x, is an exponent. Thereafter thing you must note is that there are no more terms that have the variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when you try solving various calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are very important in arithmetic and play a critical role in working out many mathematical 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 appear in the exponent of an exponential equation. Exponential equations are remarkable easy to find in daily life. There are three primary kinds of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the most convenient to solve, as we can easily set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same utilizing properties of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the described steps as the first case.
3) Equations with variable bases on each sides that is unable to be made the similar. These are the toughest to figure out, but it’s feasible using the property of the product rule. By raising both factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can determine the two new equations identical to one another and figure out the unknown variable. This article do not cover logarithm solutions, but we will tell you where to get assistance at the end of this article.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now understand how to solve any equation by ensuing 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 inside the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Thereafter, 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 initial equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at a few examples to observe how these steps work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are identical. Therefore, all you need to do is to restate the exponents and figure them out using algebra:
y+1=3y
y=½
Right away, 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 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 identical base. However, both sides are powers of two. As such, the working includes breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we figure out this expression to find the final answer:
28=22x-10
Carry out algebra to solve for x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can recheck our answer by substituting 9 for x in the original equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you use the properties of exponents, you will inturn master of these theorems, solving almost all exponential equations without issue.
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