Exponential EquationsExplanation, Solving, and Examples
In math, an exponential equation occurs when the variable appears in the exponential function. This can be a frightening topic for students, 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, proceduce to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The first step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that include 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 items to bear in mind for when trying to determine 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 (aside from the exponent)
For example, check out this equation:
y = 3x2 + 7
The most important thing you must observe is that the variable, x, is in an exponent. Thereafter thing you should notice is that there is another 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, check out this equation:
y = 2x + 5
One more time, the primary thing you should notice is that the variable, x, is an exponent. The second thing you should observe is that there are no other terms that includes any variable in them. This means that this equation IS exponential.
You will run into exponential equations when working on diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in math and play a pivotal duty in solving many mathematical problems. Therefore, it is critical to completely understand what exponential equations are and how they can be used as you move ahead in your math studies.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three major kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with different bases on each sides, but they can be created the same employing rules of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the same steps as the first event.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the most difficult to work out, but it’s feasible utilizing the property of the product rule. By increasing both factors to the same power, we can multiply the factors on each side and raise them.
Once we have done this, we can set the two new equations equal to each other and work on the unknown variable. This article do not contain logarithm solutions, but we will tell you where to get help at the closing parts of this article.
How to Solve Exponential Equations
Knowing the definition and kinds of exponential equations, we can now learn to solve 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.
Primarily, we must determine the base and exponent variables inside the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Thereafter, we can solve them using standard algebraic methods.
Lastly, we have to work on the unknown variable. Now that we have solved for the variable, we can put this value back into our initial equation to discover 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 practicality.
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 rewrite the exponents and solve through algebra:
y+1=3y
y=½
Now, we change the value of y in the specified equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated question. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. However, both sides are powers of two. As such, the working comprises of decomposing respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the final answer:
28=22x-10
Perform algebra to work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our work by substituting 9 for x in the first equation.
256=49−5=44
Continue searching for examples and questions on the internet, and if you use the rules of exponents, you will become a master of these concepts, working out almost all exponential equations without issue.
Better Your Algebra Skills with Grade Potential
Working on questions with exponential equations can be tough without help. Even though this guide goes through the fundamentals, you still might find questions or word questions that may hinder you. Or perhaps you require some extra help as logarithms come into the scenario.
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