Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation occurs when the variable shows up 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 easily.
This blog post will discuss the explanation of exponential equations, types of exponential equations, steps to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to figure out an exponential equation is knowing 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 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 only one term that has the variable in it (in addition of the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The first thing you should note is that the variable, x, is in an exponent. Thereafter thing you must not is that there is another term, 3x2, that has the variable in it – not only in an exponent. This signifies that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you must observe is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no more terms that includes any variable in them. This implies that this equation IS exponential.
You will come across exponential equations when you try solving various calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are essential in math and perform a critical role in working out many math problems. Therefore, it is crucial to fully understand what exponential equations are and how they can be utilized as you move ahead in arithmetic.
Types of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are surprisingly common in everyday life. There are three main types of exponential equations that we can solve:
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 equivalent as each other and work out for the unknown variable.
2) Equations with distinct bases on each sides, but they can be created similar using properties of the exponents. We will put a few examples below, but by changing the bases the same, you can observe the same steps as the first instance.
3) Equations with distinct bases on both sides that is unable to be made the same. These are the most difficult to figure out, but it’s attainable through the property of the product rule. By increasing two or more factors to similar power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two new equations equal to each other and work on the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get guidance at the very last of this article.
How to Solve Exponential Equations
Knowing the explanation and types of exponential equations, we can now learn to solve any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we need to ensue to solve exponential equations.
Primarily, we must identify the base and exponent variables inside the equation.
Next, we have to rewrite an exponential equation, so all terms are in common base. Then, we can solve them utilizing standard algebraic rules.
Lastly, we have to work on the unknown variable. Once we have solved for the variable, we can put this value back into our first equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at some examples to note how these process work in practicality.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are identical. Thus, all you are required to do is to restate the exponents and figure them out through algebra:
y+1=3y
y=½
So, we change the value of y in the respective 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. By itself, the solution includes breaking down 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 answer:
28=22x-10
Apply algebra to solve for x in the exponents as we did in the prior example.
8=2x-10
x=9
We can verify our workings by substituting 9 for x in the first equation.
256=49−5=44
Keep seeking for examples and questions on the internet, and if you utilize the rules of exponents, you will become a master of these theorems, solving almost all exponential equations with no issue at all.
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