Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation arises when the variable shows up in the exponential function. This can be a frightening topic for children, but with a some of instruction and practice, exponential equations can be determited quickly.
This article post will discuss the definition of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial step to work on an exponential equation is knowing 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 items to keep in mind for when trying to figure out if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is only one term that has the variable in it (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The most important thing you should notice is that the variable, x, is in an exponent. Thereafter thing you must notice is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the flipside, check out this equation:
y = 2x + 5
Yet again, the primary thing you must notice is that the variable, x, is an exponent. The second thing you must notice is that there are no other terms that includes any variable in them. This means that this equation IS exponential.
You will come upon exponential equations when you try solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are crucial in mathematics and perform a critical duty in solving many mathematical questions. Therefore, it is important to fully understand what exponential equations are and how they can be used as you move ahead in mathematics.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly common in everyday life. There are three main types 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 equivalent as each other and figure out for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created the same utilizing properties of the exponents. We will take a look at some examples below, but by changing the bases the same, you can follow the exact steps as the first instance.
3) Equations with distinct bases on each sides that cannot be made the similar. These are the most difficult to work out, but it’s attainable through the property of the product rule. By raising both factors to the same power, we can multiply the factors on both side and raise them.
Once we are done, we can set the two latest equations equal to one another and work on the unknown variable. This blog do not cover logarithm solutions, but we will let you know where to get assistance at the closing parts of this blog.
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 ensuing these easy procedures.
Steps for Solving Exponential Equations
We have three steps that we are going to ensue to work on exponential equations.
First, we must determine the base and exponent variables within the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Then, we can solve 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 original equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to note how these process work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are the same. Thus, all you have to do is to restate the exponents and work on them through algebra:
y+1=3y
y=½
Now, we substitute 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 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. But, both sides are powers of two. As such, the solution comprises of breaking down respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we solve this expression to come to the final result:
28=22x-10
Apply algebra to work out the x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can verify our work by replacing 9 for x in the first equation.
256=49−5=44
Keep looking for examples and questions over the internet, and if you utilize the rules of exponents, you will inturn master of these concepts, solving almost all exponential equations without issue.
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