Exponential EquationsExplanation, Solving, and Examples
In mathematics, an exponential equation occurs when the variable appears in the exponential function. This can be a scary topic for students, but with a some of instruction and practice, exponential equations can be determited simply.
This article post will talk about the definition of exponential equations, kinds of exponential equations, process to figure out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The first step to solving an exponential equation is understanding 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 primary items to keep in mind 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 no other term that has the variable in it (in addition of the exponent)
For example, look at this equation:
y = 3x2 + 7
The primary thing you should notice 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 means that this equation is NOT exponential.
On the flipside, check out this equation:
y = 2x + 5
One more time, the primary thing you must observe is that the variable, x, is an exponent. The second thing you must observe is that there are no other 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 compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are very important in arithmetic and play a pivotal role in solving many math problems. Thus, it is critical to fully understand what exponential equations are and how they can be utilized as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable common in daily life. There are three major types of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the easiest to work out, as we can easily set the two equations equal to each other and figure out for the unknown variable.
2) Equations with different bases on each sides, but they can be made similar utilizing rules of the exponents. We will take a look at some examples below, but by making the bases the same, you can observe the same steps as the first case.
3) Equations with different bases on both sides that is impossible to be made the same. These are the trickiest to solve, but it’s attainable utilizing 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 determine the two latest equations identical to one another and work on the unknown variable. This blog does not cover logarithm solutions, but we will let you know where to get guidance at the end of this article.
How to Solve Exponential Equations
After going through the explanation and types of exponential equations, we can now learn to work on any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
We have three steps that we are going to follow to solve exponential equations.
First, we must recognize the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Then, we can solve them through standard algebraic rules.
Lastly, we have to work on the unknown variable. Since we have figured out the variable, we can plug this value back into our initial equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at a few examples to note how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are the same. Therefore, all you are required to do is to rewrite the exponents and work on them using algebra:
y+1=3y
y=½
Right away, we change the value of y in the respective equation to corroborate that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's observe this up with a more complex problem. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a common 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 solve this expression to come to the final result:
28=22x-10
Perform algebra to work out the x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can recheck our work by replacing 9 for x in the first equation.
256=49−5=44
Keep searching for examples and problems over the internet, and if you use the laws of exponents, you will become a master of these theorems, solving most exponential equations without issue.
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