Exponential EquationsExplanation, Solving, and Examples
In math, 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 quickly.
This blog post will talk about the explanation of exponential equations, types of exponential equations, steps to solve exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining when you have one.
Definition
Exponential equations are equations that include 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 primary items to bear in mind for when trying to determine 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, take a look at this equation:
y = 3x2 + 7
The first thing you should notice is that the variable, x, is in an exponent. The second thing you should notice is that there is another term, 3x2, that has the variable in it – not only in an exponent. This implies that this equation is NOT exponential.
On the other hand, take a look at this equation:
y = 2x + 5
Once again, the first thing you must notice is that the variable, x, is an exponent. Thereafter thing you should observe is that there are no other terms that consists of any variable in them. This implies that this equation IS exponential.
You will run into exponential equations when you try solving different calculations in algebra, compound interest, exponential growth or decay, and other functions.
Exponential equations are very important in mathematics and perform a critical responsibility in figuring out many mathematical questions. Hence, it is important to completely grasp what exponential equations are and how they can be used as you move ahead in arithmetic.
Varieties of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are remarkable ordinary in daily life. There are three major 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 easily set the two equations equal to each other and figure out for the unknown variable.
2) Equations with distinct bases on each sides, but they can be created the same utilizing rules of the exponents. We will put a few examples below, but by converting the bases the equal, you can follow the same 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 using the property of the product rule. By increasing both factors to identical power, we can multiply the factors on each side and raise them.
Once we are done, we can set the two latest equations identical to each other and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get guidance at the closing parts of this blog.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now understand how to work on any equation by following these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
First, 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. Thereafter, we can solve them utilizing standard algebraic techniques.
Third, we have to work on the unknown variable. Since we have figured out the variable, we can plug this value back into our original equation to figure out the value of the other.
Examples of How to Solve 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 see that all the bases are identical. Therefore, all you have to do is to restate the exponents and work on them through 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 further complex sum. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a similar base. Despite that, both sides are powers of two. By itself, the solution includes decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we figure out this expression to conclude the ultimate answer:
28=22x-10
Apply algebra to work out the x in the exponents as we conducted in the previous example.
8=2x-10
x=9
We can recheck our answer by substituting 9 for x in the first equation.
256=49−5=44
Continue searching for examples and problems on the internet, and if you utilize the laws of exponents, you will inturn master of these theorems, solving almost all exponential equations with no issue at all.
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Solving problems with exponential equations can be tough without help. Although this guide take you through the fundamentals, you still may find questions or word problems that make you stumble. Or maybe you require some additional guidance as logarithms come into play.
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