Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a frightening topic for students, but with a some of instruction and practice, exponential equations can be determited simply.
This blog post will talk about the definition of exponential equations, kinds of exponential equations, proceduce to figure out exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The primary step to work on an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that consist of 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 things to look for when attempting to establish 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 (in addition of the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you must observe 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 implies that this equation is NOT exponential.
On the flipside, look at 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 notice is that there are no other value that consists of any variable in them. This implies that this equation IS exponential.
You will come across exponential equations when working on various calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are very important in mathematics and perform a central responsibility in working out many computational problems. Thus, it is important to completely understand what exponential equations are and how they can be used as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly common in daily life. There are three main kinds of exponential equations that we can figure out:
1) Equations with identical bases on both sides. This is the easiest to work out, as we can easily set the two equations same as each other and figure out for the unknown variable.
2) Equations with different bases on each sides, but they can be made the same using properties of the exponents. We will take a look at some examples below, but by making the bases the same, you can observe the exact steps as the first instance.
3) Equations with distinct bases on each sides that is impossible to be made the same. These are the trickiest to figure out, but it’s possible 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 identical to each other and figure out the unknown variable. This article does not include logarithm solutions, but we will tell you where to get help at the closing parts of this blog.
How to Solve Exponential Equations
Knowing 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
Remember these three steps that we need to ensue to work on exponential equations.
Primarily, we must identify the base and exponent variables in the equation.
Next, we are required to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them utilizing standard algebraic techniques.
Lastly, we have to work on the unknown variable. Now that 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 note how these process work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can see that all the bases are identical. Thus, all you need to do is to rewrite the exponents and solve using algebra:
y+1=3y
y=½
Right away, we substitute the value of y in the respective 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 complex sum. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a similar base. Despite that, both sides are powers of two. In essence, the solution includes breaking down respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve 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 recheck our workings by replacing 9 for x in the original equation.
256=49−5=44
Keep looking for examples and questions on the internet, and if you use the properties of exponents, you will turn into a master of these concepts, figuring out almost all exponential equations with no issue at all.
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Working on questions with exponential equations can be tricky without support. Although this guide take you through the basics, you still may find questions or word questions that might stumble you. Or maybe you desire some additional assistance as logarithms come into the scenario.
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