Exponential EquationsDefinition, 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 some of instruction and practice, exponential equations can be solved 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 answers. Let's get started!
What Is an Exponential Equation?
The initial step to solving an exponential equation is understanding when you are working with 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 key things to keep in mind 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 no other term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you should notice 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 – just not in an exponent. This means that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Once again, the first thing you should notice is that the variable, x, is an exponent. The second thing you should note is that there are no more value that consists of any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when working on different calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are crucial in arithmetic and perform a critical role in figuring out many math questions. Hence, it is critical to completely understand what exponential equations are and how they can be utilized as you go ahead in mathematics.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three primary kinds of exponential equations that we can work out:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can simply set the two equations equal to each other and work out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be created similar using rules of the exponents. We will take a look at some examples below, but by converting the bases the equal, you can observe the described steps as the first instance.
3) Equations with variable bases on both sides that is impossible to be made the same. These are the most difficult to solve, but it’s possible utilizing the property of the product rule. By raising two or more factors to similar power, we can multiply the factors on each side and raise them.
Once we are done, we can determine the two new equations identical to each other and solve for the unknown variable. This blog do not cover logarithm solutions, but we will tell you where to get assistance at the closing parts of this blog.
How to Solve Exponential Equations
After going through the definition and types of exponential equations, we can now learn to work on any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we are going to ensue to solve exponential equations.
First, we must identify the base and exponent variables within the equation.
Next, we need to rewrite an exponential equation, so all terms have a common base. Thereafter, we can solve them through standard algebraic techniques.
Lastly, we have to solve for the unknown variable. Since we have solved for the variable, we can plug this value back into our first equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to see how these process work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can observe that both bases are identical. Thus, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
So, we change 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 sum. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. Despite that, both sides are powers of two. As such, the working 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 find the final result:
28=22x-10
Carry out algebra to solve for x in the exponents as we did in the prior example.
8=2x-10
x=9
We can double-check our answer by replacing 9 for x in the original equation.
256=49−5=44
Keep looking for examples and questions on the internet, and if you use the laws of exponents, you will turn into a master of these concepts, solving most exponential equations with no issue at all.
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