Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation arises when the variable shows up in the exponential function. This can be a scary topic for students, but with a bit of direction and practice, exponential equations can be determited simply.
This blog post will talk about the definition of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with answers. Let's get started!
What Is an Exponential Equation?
The primary step to work on 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 instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary things to look for when you seek to determine 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 (besides the exponent)
For example, look at 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 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 other hand, take a look at this equation:
y = 2x + 5
Once again, the first thing you should note is that the variable, x, is an exponent. The second thing you should notice is that there are no other terms that consists of any variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when solving various calculations in exponential growth, algebra, compound interest or decay, and various distinct functions.
Exponential equations are very important in mathematics and perform a critical responsibility in solving many mathematical problems. Thus, it is critical to completely understand what exponential equations are and how they can be used as you progress in arithmetic.
Varieties of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are remarkable ordinary in daily life. There are three primary types of exponential equations that we can solve:
1) Equations with identical bases on both sides. This is the simplest to solve, as we can easily set the two equations same as each other and solve for the unknown variable.
2) Equations with distinct bases on each sides, but they can be created similar utilizing rules of the exponents. We will show some examples below, but by converting the bases the same, you can follow the same steps as the first event.
3) Equations with variable bases on both sides that cannot be made the similar. These are the most difficult to figure out, but it’s feasible utilizing the property of the product rule. By raising two or more factors to identical 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 each other and figure out the unknown variable. This blog do not include logarithm solutions, but we will let you know where to get help at the closing parts of this article.
How to Solve Exponential Equations
After going through the explanation and kinds of exponential equations, we can now move on to how to solve any equation by ensuing these simple procedures.
Steps for Solving Exponential Equations
Remember these three steps that we need to follow to solve exponential equations.
Primarily, we must identify 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 methods.
Lastly, we have to figure out the unknown variable. Once we have solved for the variable, we can put this value back into our first equation to discover the value of the other.
Examples of How to Work on Exponential Equations
Let's check out a few examples to note how these steps work in practice.
Let’s start, we will work on the following example:
7y + 1 = 73y
We can see that both bases are identical. Hence, all you have to do is to restate the exponents and figure them out using algebra:
y+1=3y
y=½
So, we substitute 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 follow this up with a further complicated sum. Let's work on this expression:
256=4x−5
As you can see, the sides of the equation does not share a common base. But, both sides are powers of two. By itself, the working includes decomposing respectively the 4 and the 256, and we can alter the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the final result:
28=22x-10
Carry out algebra to solve for x in the exponents as we did in the last example.
8=2x-10
x=9
We can double-check our work by altering 9 for x in the first equation.
256=49−5=44
Keep looking for examples and problems over the internet, and if you utilize the properties of exponents, you will turn into a master of these theorems, solving almost all exponential equations with no issue at all.
Better Your Algebra Skills with Grade Potential
Working on problems with exponential equations can be difficult in absence help. Although this guide goes through the fundamentals, you still may find questions or word questions that may hinder you. Or perhaps you need some additional assistance as logarithms come into the scene.
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