Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation arises when the variable shows up in the exponential function. This can be a terrifying topic for students, but with a bit of direction and practice, exponential equations can be worked out simply.
This article post will talk about the definition of exponential equations, kinds of exponential equations, steps to solve exponential equations, and examples with solutions. Let's get right to it!
What Is an Exponential Equation?
The initial step to solving an exponential equation is understanding when you are working with 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 things to keep in mind for when you seek to establish if an equation is exponential:
1. The variable is in an exponent (signifying 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, look at this equation:
y = 3x2 + 7
The most important thing you must note is that the variable, x, is in an exponent. The second thing you must notice is that there is one more term, 3x2, that has the variable in it – just not 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 should notice is that the variable, x, is an exponent. The second thing you must observe is that there are no other value that consists of any variable in them. This means that this equation IS exponential.
You will come upon exponential equations when working on different calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are essential in mathematics and play a pivotal responsibility in solving many math problems. Therefore, it is important to fully grasp what exponential equations are and how they can be used as you move ahead in arithmetic.
Varieties of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are amazingly easy to find in daily life. There are three primary kinds of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the most convenient 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 similar using properties of the exponents. We will put a few examples below, but by making the bases the same, you can observe the same steps as the first event.
3) Equations with variable bases on both sides that is impossible to be made the similar. These are the toughest to solve, but it’s feasible utilizing the property of the product rule. By raising 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 set the two new equations equal to one another and figure out the unknown variable. This blog does not cover logarithm solutions, but we will tell you where to get help at the closing parts of this article.
How to Solve Exponential Equations
From the explanation and kinds of exponential equations, we can now learn to solve any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
There are three steps that we are going to ensue to solve exponential equations.
First, we must recognize the base and exponent variables within the equation.
Second, we have to rewrite an exponential equation, so all terms have a common base. Then, we can solve them through standard algebraic rules.
Lastly, we have to figure out the unknown variable. Once we have figured out the variable, we can put this value back into our initial equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's take a loot at some examples to see how these steps work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can observe that all the bases are the same. Therefore, all you need to do is to restate the exponents and solve through algebra:
y+1=3y
y=½
Right away, we replace the value of y in the given equation to support that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex problem. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. However, both sides are powers of two. By itself, the working consists of breaking down both the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we work on this expression to find the final answer:
28=22x-10
Perform algebra to solve for x in the exponents as we did in the previous example.
8=2x-10
x=9
We can recheck our work by altering 9 for x in the original equation.
256=49−5=44
Keep searching for examples and problems on the internet, and if you use the rules of exponents, you will inturn master of these concepts, solving most exponential equations with no issue at all.
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Working on problems with exponential equations can be tricky with lack of support. Although this guide covers the basics, you still might face questions or word questions that make you stumble. Or perhaps you require some additional assistance as logarithms come into the scene.
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