Exponential EquationsExplanation, Workings, and Examples
In mathematics, an exponential equation arises when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a bit of direction and practice, exponential equations can be determited easily.
This article post will discuss the definition of exponential equations, types 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 figure out an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that have 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 major items to keep in mind for when you seek 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 (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you should note is that the variable, x, is in an exponent. Thereafter 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
One more time, the first thing you should notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more value that have the variable in them. This means that this equation IS exponential.
You will run into exponential equations when working on different calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are crucial in mathematics and perform a critical responsibility in solving many math questions. Hence, it is crucial to completely grasp what exponential equations are and how they can be utilized as you go ahead in mathematics.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly easy to find in everyday life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient to solve, as we can simply set the two equations same as each other and figure out for the unknown variable.
2) Equations with distinct bases on both sides, but they can be created the same utilizing properties of the exponents. We will show some examples below, but by converting the bases the same, you can follow the same steps as the first instance.
3) Equations with variable 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 both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two latest equations identical to one another and solve for the unknown variable. This article do not cover logarithm solutions, but we will tell you where to get help at the end of this blog.
How to Solve Exponential Equations
Knowing the explanation and kinds of exponential equations, we can now learn to work on any equation by following these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to ensue to solve exponential equations.
First, we must determine the base and exponent variables in the equation.
Second, we are required to rewrite an exponential equation, so all terms are in common base. Subsequently, we can work on them through standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Since we have figured out the variable, we can plug this value back into our original equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to observe how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that both bases are the same. Thus, all you have to do is to rewrite the exponents and work on them through algebra:
y+1=3y
y=½
Now, we replace the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complicated problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation do not share a similar base. However, both sides are powers of two. In essence, the solution comprises of decomposing both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we figure out this expression to conclude the final answer:
28=22x-10
Apply algebra to figure out x in the exponents as we did in the prior example.
8=2x-10
x=9
We can verify our work by altering 9 for x in the original equation.
256=49−5=44
Continue looking for examples and problems on the internet, and if you use the rules of exponents, you will turn into a master of these theorems, working out almost all exponential equations with no issue at all.
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