Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation occurs when the variable shows up in the exponential function. This can be a terrifying topic for children, but with a bit of direction and practice, exponential equations can be determited simply.
This article post will talk about the explanation of exponential equations, types of exponential equations, process to solve exponential equations, and examples with solutions. Let's began!
What Is an Exponential Equation?
The initial step to work on an exponential equation is understanding when you are working with 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 key items to look for when attempting 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 (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must observe is that the variable, x, is in an exponent. The second thing you must observe is that there is another term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the other hand, take a look at this equation:
y = 2x + 5
One more time, the primary thing you should notice is that the variable, x, is an exponent. The second thing you must observe is that there are no other value that includes any variable in them. This means that this equation IS exponential.
You will come upon exponential equations when solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in mathematics and play a critical role in working out many mathematical questions. Therefore, it is important to fully grasp what exponential equations are and how they can be used as you progress in mathematics.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable easy to find in everyday life. There are three primary kinds 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 simply set the two equations equal to each other and figure out for the unknown variable.
2) Equations with distinct bases on both 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 equal, you can observe the exact steps as the first event.
3) Equations with variable bases on both sides that is unable to be made the similar. These are the most difficult to solve, but it’s attainable using the property of the product rule. By raising both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two new equations equal to one another and solve for the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get assistance at the very last of this blog.
How to Solve Exponential Equations
From the definition and types of exponential equations, we can now understand how to solve any equation by ensuing these easy procedures.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
First, we must determine the base and exponent variables in the equation.
Next, we need to rewrite an exponential equation, so all terms are in common base. Subsequently, we can solve them utilizing standard algebraic methods.
Lastly, we have to solve for the unknown variable. Since we have figured out the variable, we can put this value back into our initial equation to figure out the value of the other.
Examples of How to Work on Exponential Equations
Let's look at a few examples to note how these process work in practicality.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that both bases are the same. Thus, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
Now, 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 further complex sum. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. Despite that, both sides are powers of two. As such, the working comprises of decomposing respectively the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we solve this expression to conclude the final answer:
28=22x-10
Apply algebra to solve for x in the exponents as we conducted in the prior example.
8=2x-10
x=9
We can recheck our workings by replacing 9 for x in the original equation.
256=49−5=44
Keep seeking for examples and problems online, and if you use the properties of exponents, you will inturn master of these concepts, solving most exponential equations with no issue at all.
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