Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation takes place when the variable appears in the exponential function. This can be a scary topic for kids, but with a bit of instruction and practice, exponential equations can be solved simply.
This article post will talk about the explanation of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial 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 example, 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 attempting 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 no other term that has the variable in it (besides the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The primary thing you must note is that the variable, x, is in an exponent. The second thing you should notice is that there is one more term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you should notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no other terms that includes any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are crucial in mathematics and play a pivotal responsibility in working out many mathematical questions. Thus, it is critical to fully grasp what exponential equations are and how they can be used as you move ahead in mathematics.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in everyday life. There are three major types of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can simply 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 made similar utilizing properties of the exponents. We will take a look at some examples below, but by changing the bases the same, you can observe the described steps as the first instance.
3) Equations with variable bases on each sides that cannot be made the same. These are the toughest to figure out, but it’s possible utilizing the property of the product rule. By increasing both factors to the same power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two new equations identical to each other and work on the unknown variable. This article does not cover logarithm solutions, but we will tell you where to get help at the closing parts of this article.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now move on to how to solve any equation by following these easy steps.
Steps for Solving Exponential Equations
There are three steps that we are required to ensue to work on exponential equations.
First, we must identify the base and exponent variables within the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Subsequently, we can solve them using standard algebraic methods.
Third, we have to figure out the unknown variable. Once we have solved for the variable, we can put this value back into our original equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at a few examples to observe how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can observe that both bases are the same. Thus, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
Right away, we change 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 complicated question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation do not share a common base. But, both sides are powers of two. As such, the working includes breaking down both 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 final answer:
28=22x-10
Carry out algebra to work out the x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can verify our work by substituting 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems online, and if you utilize the rules of exponents, you will inturn master of these theorems, figuring out almost all exponential equations without issue.
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