Exponential EquationsExplanation, Solving, and Examples
In arithmetic, an exponential equation occurs when the variable appears in the exponential function. This can be a scary topic for students, 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, steps to work out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The initial step to figure out an exponential equation is knowing when you have 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 look for when you seek to determine 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 (aside from the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The first thing you should observe is that the variable, x, is in an exponent. Thereafter thing you should not 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 flipside, take a look at this equation:
y = 2x + 5
One more time, the first thing you must notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more terms that consists of any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving different calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in arithmetic and play a pivotal role in figuring out many mathematical problems. Therefore, it is crucial to completely grasp what exponential equations are and how they can be used as you go ahead in your math studies.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly ordinary in daily life. There are three primary types of exponential equations that we can figure out:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can easily set the two equations equivalent as each other and figure out for the unknown variable.
2) Equations with dissimilar bases on each sides, but they can be created the same utilizing properties of the exponents. We will take a look at some examples below, but by converting the bases the equal, you can observe the same steps as the first instance.
3) Equations with different bases on each sides that cannot be made the similar. These are the most difficult to figure out, but it’s possible using the property of the product rule. By raising both factors to similar power, we can multiply the factors on both side and raise them.
Once we are done, we can resolute the two new equations equal to each other and solve for the unknown variable. This article do not cover logarithm solutions, but we will let you know where to get assistance at the end of this blog.
How to Solve Exponential Equations
Knowing the definition and types of exponential equations, we can now move on to how to work on any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
There are three steps that we need to ensue to solve exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Then, we can solve them utilizing standard algebraic rules.
Lastly, we have to figure out the unknown variable. Once we have figured out 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 look at some examples to see how these procedures work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are the same. Hence, all you have to do is to restate the exponents and work on them utilizing algebra:
y+1=3y
y=½
Right away, we change the value of y in the respective equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complicated sum. Let's solve this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. But, 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 work on this expression to find the ultimate result:
28=22x-10
Apply algebra to figure out x in the exponents as we did in the previous example.
8=2x-10
x=9
We can verify our work by replacing 9 for x in the first equation.
256=49−5=44
Keep seeking for examples and problems over the internet, and if you use the properties of exponents, you will turn into a master of these concepts, working out almost all exponential equations without issue.
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