Exponential EquationsExplanation, Workings, and Examples
In math, an exponential equation occurs 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 blog post will talk about the explanation of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The initial step to work on an exponential equation is understanding 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 key things to keep in mind for when you seek 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, check out this equation:
y = 3x2 + 7
The primary thing you should observe is that the variable, x, is in an exponent. Thereafter thing you should observe is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you should 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 upon exponential equations when solving different calculations in algebra, compound interest, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and perform a pivotal duty in figuring out many computational problems. Thus, it is crucial to fully grasp what exponential equations are and how they can be utilized as you move ahead in mathematics.
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 types of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the easiest to solve, as we can simply set the two equations equivalent as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be created the same utilizing properties of the exponents. We will show some examples below, but by changing the bases the same, you can follow the exact steps as the first event.
3) Equations with distinct bases on both sides that is unable to be made the similar. These are the toughest to figure out, but it’s feasible using the property of the product rule. By increasing two or more factors to identical power, we can multiply the factors on both side and raise them.
Once we are done, we can set the two latest equations equal to one another and solve for the unknown variable. This blog does not contain logarithm solutions, but we will let you know where to get assistance at the closing parts of this blog.
How to Solve Exponential Equations
After going through the explanation and kinds 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 are required to ensue to solve exponential equations.
First, we must identify the base and exponent variables inside the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Then, we can work on them utilizing standard algebraic methods.
Third, we have to figure out the unknown variable. Since we have solved for the variable, we can plug this value back into our first equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's check out some examples to note how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can observe that all the bases are identical. Therefore, all you need to do is to restate the exponents and work on them using algebra:
y+1=3y
y=½
Right away, we change 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 observe this up with a more complicated problem. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a identical base. Despite that, both sides are powers of two. By itself, the solution includes breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the ultimate answer:
28=22x-10
Carry out algebra to figure out x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can double-check our answer by replacing 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and questions over the internet, and if you use the properties of exponents, you will turn into a master of these concepts, working out most exponential equations with no issue at all.
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