Exponential EquationsExplanation, Workings, and Examples
In arithmetic, an exponential equation arises when the variable shows up in the exponential function. This can be a scary topic for students, but with a some of direction and practice, exponential equations can be worked out quickly.
This blog post will talk about the explanation of exponential equations, kinds of exponential equations, steps to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that have 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 major things to keep in mind for when attempting 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 no other term that has the variable in it (aside from the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you should notice 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 – not only in an exponent. This implies that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
One more time, the first thing you should note is that the variable, x, is an exponent. The second thing you should observe is that there are no other value that have the variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when working on diverse calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are very important in mathematics and perform a critical duty in figuring out many computational problems. Thus, it is important to completely grasp what exponential equations are and how they can be utilized as you progress in arithmetic.
Kinds of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable easy to find in everyday life. There are three main 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 easily set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with different bases on each sides, but they can be created similar employing properties of the exponents. We will take a look at some examples below, but by converting the bases the equal, you can observe the described steps as the first instance.
3) Equations with distinct bases on both sides that is unable to be made the similar. These are the trickiest to solve, but it’s attainable utilizing the property of the product rule. By raising both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can set the two latest equations equal to one another and work on the unknown variable. This article 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
After going through the definition and types of exponential equations, we can now learn to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are required to ensue to work on exponential equations.
First, we must determine the base and exponent variables within the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them using standard algebraic rules.
Third, we have to figure out the unknown variable. Once we have solved for the variable, we can put this value back into our initial equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's check out some examples to observe how these steps work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can observe that both bases are identical. Thus, all you need to do is to rewrite the exponents and work on them through algebra:
y+1=3y
y=½
So, we substitute the value of y in the specified equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complicated question. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. But, both sides are powers of two. By itself, the working comprises of 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 ultimate result:
28=22x-10
Apply algebra to solve for x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can verify our workings by replacing 9 for x in the original equation.
256=49−5=44
Continue looking for examples and questions online, and if you use the properties of exponents, you will become a master of these concepts, solving almost all exponential equations without issue.
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