Exponential EquationsDefinition, Workings, and Examples
In mathematics, an exponential equation occurs when the variable appears in the exponential function. This can be a frightening topic for children, but with a bit of instruction and practice, exponential equations can be determited quickly.
This article post will talk about the explanation of exponential equations, types of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The first step to figure out an exponential equation is determining when you have one.
Definition
Exponential equations are equations that consist of 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 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 (besides the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you must note is that the variable, x, is in an exponent. The second thing you must not is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This signifies that this equation is NOT exponential.
On the other hand, look at this equation:
y = 2x + 5
Yet again, the primary thing you should observe is that the variable, x, is an exponent. The second thing you should note is that there are no other terms that includes any variable in them. This signifies that this equation IS exponential.
You will come across exponential equations when solving various calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are very important in mathematics and play a central role in solving many computational questions. Thus, it is critical to fully understand what exponential equations are and how they can be used as you move ahead in arithmetic.
Kinds of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are amazingly common in everyday life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the simplest to work out, as we can easily set the two equations same as each other and work out for the unknown variable.
2) Equations with distinct bases on each sides, but they can be made the same utilizing properties of the exponents. We will take a look at some examples below, but by changing the bases the equal, you can follow the same steps as the first instance.
3) Equations with variable bases on both sides that is unable to be made the similar. These are the most difficult to work out, but it’s feasible utilizing the property of the product rule. By increasing both factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can set the two new equations identical to one another and solve for the unknown variable. This article does not include logarithm solutions, but we will tell you where to get help at the very last of this article.
How to Solve Exponential Equations
After going through the explanation and kinds of exponential equations, we can now learn 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 within the equation.
Next, we have to rewrite an exponential equation, so all terms have a common base. Subsequently, we can solve them through standard algebraic rules.
Third, we have to figure out the unknown variable. Now that we have solved for the variable, we can plug this value back into our original equation to discover the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to note how these procedures work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can notice that both bases are the same. Therefore, all you have to do is to rewrite the exponents and figure them out using algebra:
y+1=3y
y=½
So, 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 follow this up with a further complicated question. Let's work on this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a common base. Despite that, both sides are powers of two. As such, the solution includes decomposing respectively 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 ultimate answer:
28=22x-10
Carry out algebra to work out the x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can double-check our workings by replacing 9 for x in the original equation.
256=49−5=44
Continue looking for examples and problems online, and if you utilize the rules of exponents, you will become a master of these concepts, figuring out most exponential equations with no issue at all.
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Working on problems with exponential equations can be tricky in absence help. Although this guide goes through the essentials, you still might face questions or word problems that might stumble you. Or maybe you require some additional assistance as logarithms come into play.
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