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 students, but with a bit of instruction and practice, exponential equations can be worked out easily.
This blog post will talk about the definition of exponential equations, types of exponential equations, steps to figure out exponential equations, and examples with answers. Let's began!
What Is an Exponential Equation?
The primary step to figure out an exponential equation is knowing when you have one.
Definition
Exponential equations are equations that include 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 major things to bear in mind for when trying 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 (in addition of the exponent)
For example, take a 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 notice is that there is additional term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the flipside, take a look at this equation:
y = 2x + 5
Once again, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you must note is that there are no other terms that have the variable in them. This implies that this equation IS exponential.
You will come across exponential equations when solving diverse calculations in exponential growth, algebra, compound interest or decay, and other functions.
Exponential equations are very important in arithmetic and play a pivotal role in figuring out many math questions. Hence, it is important to fully understand what exponential equations are and how they can be used as you progress in arithmetic.
Types of Exponential Equations
Variables occur in the exponent of an exponential equation. Exponential equations are surprisingly easy to find 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 solve for the unknown variable.
2) Equations with distinct bases on both sides, but they can be made similar utilizing properties of the exponents. We will put a few examples below, but by making the bases the equal, you can observe the same steps as the first case.
3) Equations with variable bases on each sides that is unable to be made the similar. These are the most difficult to work out, but it’s feasible using the property of the product rule. By raising both factors to the same power, we can multiply the factors on both side and raise them.
Once we have done this, we can set the two new equations identical to each other and solve for the unknown variable. This blog does not cover logarithm solutions, but we will tell you where to get help at the very last of this blog.
How to Solve Exponential Equations
Knowing the explanation and kinds 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 are required to ensue 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. Thereafter, we can work on them using standard algebraic techniques.
Third, we have to work on the unknown variable. Now that we have figured out 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 some examples to see how these steps work in practicality.
First, we will solve the following example:
7y + 1 = 73y
We can notice that both bases are identical. Therefore, all you are required to do is to restate the exponents and work on them using algebra:
y+1=3y
y=½
Right away, we substitute 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 more complicated problem. Let's figure out this expression:
256=4x−5
As you can see, the sides of the equation do not share a common base. Despite that, both sides are powers of two. In essence, the solution includes breaking down both the 4 and the 256, and we can replace the terms as follows:
28=22(x-5)
Now we figure out this expression to come to the final answer:
28=22x-10
Apply algebra to work out the x in the exponents as we performed in the previous example.
8=2x-10
x=9
We can recheck our work by substituting 9 for x in the first equation.
256=49−5=44
Continue seeking for examples and problems online, and if you utilize the laws of exponents, you will inturn master of these theorems, solving most exponential equations with no issue at all.
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