Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation takes place when the variable appears in the exponential function. This can be a terrifying topic for students, but with a bit of direction and practice, exponential equations can be solved simply.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, process to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The first step to work on an exponential equation is knowing when you are working with one.
Definition
Exponential equations are equations that include 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 items to look 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 (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you should observe is that the variable, x, is in an exponent. Thereafter thing you must not is that there is another term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the contrary, check out this equation:
y = 2x + 5
Yet again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you must note is that there are no more terms that consists of any variable in them. This means that this equation IS exponential.
You will come across exponential equations when working on diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in arithmetic and play a critical duty in figuring out many mathematical problems. Therefore, it is critical to fully grasp what exponential equations are and how they can be used as you go ahead in mathematics.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable ordinary in everyday life. There are three primary types of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the easiest to solve, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be made similar utilizing rules of the exponents. We will put a few examples below, but by changing the bases the equal, you can observe the exact steps as the first instance.
3) Equations with distinct bases on both sides that is impossible to be made the same. These are the most difficult to solve, but it’s feasible using the property of the product rule. By increasing two or more factors to similar power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two latest equations identical to each other and figure out the unknown variable. This blog does not cover logarithm solutions, but we will tell you where to get guidance at the very last of this blog.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now understand how to solve any equation by following these easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are going to follow to solve exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Second, we need to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them through standard algebraic rules.
Lastly, we have to solve for the unknown variable. Once we have figured out the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at a few examples to see how these procedures work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Therefore, all you have to do is to rewrite the exponents and work on them utilizing algebra:
y+1=3y
y=½
Now, we change the value of y in the specified 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 complicated question. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. However, both sides are powers of two. By itself, the working comprises of decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to conclude the ultimate result:
28=22x-10
Perform algebra to figure out x in the exponents as we did in the last example.
8=2x-10
x=9
We can double-check our answer by replacing 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems online, and if you utilize the properties of exponents, you will inturn master of these theorems, solving most exponential equations with no issue at all.
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