Exponential EquationsDefinition, Workings, and Examples
In arithmetic, an exponential equation takes place when the variable appears in the exponential function. This can be a frightening topic for children, but with a bit of instruction and practice, exponential equations can be worked out quickly.
This article post will discuss the explanation of exponential equations, kinds of exponential equations, steps to figure out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The initial 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 key items to look for when you seek 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 (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The most important thing you should observe is that the variable, x, is in an exponent. The second thing you must observe is that there is one more term, 3x2, that has the variable in it – not only in an exponent. This means that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
One more time, the primary thing you should note is that the variable, x, is an exponent. The second thing you must notice is that there are no other terms that have the variable in them. This means 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 essential in math and play a pivotal duty in working out many math questions. Thus, it is crucial to fully grasp what exponential equations are and how they can be used as you move ahead in arithmetic.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly ordinary in everyday life. There are three major types of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the easiest to work out, as we can simply set the two equations equal to each other and solve for the unknown variable.
2) Equations with different bases on both sides, but they can be created similar employing rules of the exponents. We will put a few examples below, but by converting the bases the same, you can observe the described steps as the first instance.
3) Equations with distinct bases on both sides that is impossible to be made the similar. These are the most difficult to solve, but it’s feasible through the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two new equations equal to one another and figure out the unknown variable. This blog do not cover logarithm solutions, but we will let you know where to get help at the end of this blog.
How to Solve Exponential Equations
After going through the definition and types of exponential equations, we can now move on to how to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
There are three steps that we need to follow to solve exponential equations.
Primarily, we must determine the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can solve them through standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Once we have figured out the variable, we can plug this value back into our first equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at some examples to note how these steps work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are the same. Thus, all you need to do is to restate the exponents and solve through algebra:
y+1=3y
y=½
Now, we substitute 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 observe this up with a more complex sum. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a identical base. But, both sides are powers of two. By itself, the solution consists of breaking down respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we work on this expression to find the ultimate result:
28=22x-10
Carry out algebra to work out the x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can double-check our work by altering 9 for x in the first equation.
256=49−5=44
Keep searching for examples and questions online, and if you use the laws of exponents, you will turn into a master of these concepts, figuring out almost all exponential equations without issue.
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Solving questions with exponential equations can be tricky in absence help. Even though this guide take you through the essentials, you still might encounter questions or word questions that might stumble you. Or maybe you need some additional assistance as logarithms come into the scenario.
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