Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a scary topic for students, but with a bit of direction and practice, exponential equations can be worked out quickly.
This article post will talk about the explanation of exponential equations, kinds of exponential equations, steps to work out exponential equations, and examples with solutions. Let's began!
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 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 major things to look for when trying 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 only one term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The primary thing you must notice is that the variable, x, is in an exponent. Thereafter thing you must notice is that there is additional term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Once again, the first thing you should note is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other terms that consists of any variable in them. This means that this equation IS exponential.
You will run into exponential equations when working on various calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are crucial in math and play a critical duty in figuring out many computational problems. Hence, it is critical to fully grasp what exponential equations are and how they can be utilized as you progress in arithmetic.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly 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 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 distinct bases on each sides, but they can be created similar utilizing rules of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the same steps as the first event.
3) Equations with variable bases on both sides that is unable to be made the similar. These are the trickiest to work out, but it’s possible using the property of the product rule. By raising two or more factors to the same power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two latest equations identical to one another and work on the unknown variable. This blog do not contain logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
From 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
There are three steps that we need to follow to work on exponential equations.
Primarily, we must determine the base and exponent variables inside the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them through standard algebraic rules.
Third, we have to solve for the unknown variable. Now that we have figured out the variable, we can plug this value back into our initial equation to find the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to observe how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that both bases are the same. Therefore, all you have to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Right away, we substitute 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 further complex question. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a similar base. But, both sides are powers of two. As such, the working consists 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 come to the ultimate answer:
28=22x-10
Perform algebra to figure out x in the exponents as we did in the prior example.
8=2x-10
x=9
We can double-check our workings by replacing 9 for x in the original equation.
256=49−5=44
Keep searching for examples and questions over the internet, and if you utilize the laws of exponents, you will inturn master of these theorems, figuring out most exponential equations with no issue at all.
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Working on problems with exponential equations can be tough in absence support. Even though this guide covers the fundamentals, you still may face questions or word problems that make you stumble. Or maybe you require some extra help as logarithms come into play.
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