Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation occurs when the variable shows up in the exponential function. This can be a scary topic for children, but with a some of direction and practice, exponential equations can be solved easily.
This blog post will talk about the explanation of exponential equations, types of exponential equations, steps to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The primary step to solving an exponential equation is knowing when you have 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 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 no other term that has the variable in it (besides 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. The second thing you should observe is that there is another term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the primary thing you should note is that the variable, x, is an exponent. Thereafter thing you should notice is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving diverse calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in math and perform a central responsibility in figuring out many computational problems. Therefore, it is critical to completely grasp what exponential equations are and how they can be utilized as you progress in arithmetic.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three major types of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the most convenient to solve, as we can easily set the two equations equal to each other and figure out for the unknown variable.
2) Equations with different bases on both sides, but they can be made the same using properties of the exponents. We will put a few examples below, but by converting the bases the same, you can observe the same steps as the first case.
3) Equations with variable bases on both sides that is impossible to be made the similar. These are the most difficult to work out, but it’s feasible 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 have done this, we can set the two new equations equal to one another and work on the unknown variable. This article do not contain logarithm solutions, but we will let you know where to get guidance 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 ensuing these simple steps.
Steps for Solving Exponential Equations
There are three steps that we are going to ensue to work on exponential equations.
Primarily, 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. Thereafter, we can solve them utilizing standard algebraic rules.
Third, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our first equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these steps work in practice.
Let’s start, we will solve the following example:
7y + 1 = 73y
We can notice that both bases are the same. Thus, all you need to do is to restate the exponents and figure them out utilizing algebra:
y+1=3y
y=½
Right away, we change 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 more complicated problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. But, both sides are powers of two. As such, the solution consists of breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to come to the final answer:
28=22x-10
Perform algebra to figure out x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can verify our work by substituting 9 for x in the original equation.
256=49−5=44
Continue looking for examples and questions on the internet, and if you utilize the laws of exponents, you will turn into a master of these theorems, figuring out most exponential equations with no issue at all.
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