Exponential EquationsDefinition, Solving, and Examples
In mathematics, an exponential equation occurs when the variable appears in the exponential function. This can be a scary topic for students, but with a some of direction and practice, exponential equations can be worked out quickly.
This blog post will talk about the explanation of exponential equations, kinds of exponential equations, proceduce to solve exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The initial step to figure out 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 keep in mind for when attempting to determine if an equation is exponential:
1. The variable is in an exponent (meaning it is raised to a power)
2. There is no other term that has the variable in it (aside from the exponent)
For example, take a look at this equation:
y = 3x2 + 7
The most important thing you must notice is that the variable, x, is in an exponent. Thereafter thing you must observe 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 contrary, look at this equation:
y = 2x + 5
One more time, the primary thing you should observe is that the variable, x, is an exponent. The second thing you should observe 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 various distinct functions.
Exponential equations are essential in mathematics and play a critical role in working out many math questions. Therefore, it is critical to completely grasp what exponential equations are and how they can be utilized as you progress in your math studies.
Kinds of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are remarkable 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 simplest to work out, as we can easily set the two equations equivalent as each other and work out for the unknown variable.
2) Equations with different bases on each sides, but they can be created similar employing properties of the exponents. We will put a few examples below, but by converting the bases the same, you can observe the same steps as the first case.
3) Equations with variable bases on both sides that cannot be made the similar. These are the trickiest to solve, but it’s feasible through the property of the product rule. By increasing two or more factors to similar power, we can multiply the factors on each side and raise them.
Once we have done this, we can determine the two new equations equal to each other and solve for the unknown variable. This blog does not cover logarithm solutions, but we will tell you where to get guidance at the closing parts of this blog.
How to Solve Exponential Equations
From the definition and kinds of exponential equations, we can now understand how to solve any equation by ensuing these simple steps.
Steps for Solving Exponential Equations
There are three steps that we are required to follow to solve exponential equations.
First, we must determine the base and exponent variables inside the equation.
Second, we are required to rewrite an exponential equation, so all terms have a common base. Subsequently, we can work on them through standard algebraic rules.
Lastly, we have to work on the unknown variable. Once we have figured out the variable, we can put this value back into our original equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's take a loot at some examples to note how these steps work in practice.
First, we will work on the following example:
7y + 1 = 73y
We can see that both bases are identical. Therefore, all you need to do is to restate the exponents and figure them out through algebra:
y+1=3y
y=½
Now, we substitute the value of y in the specified 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 complicated question. Let's figure out this expression:
256=4x−5
As you have noticed, the sides of the equation does not share a identical base. Despite that, both sides are powers of two. As such, the solution consists of breaking down both the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we figure out this expression to find the ultimate result:
28=22x-10
Apply algebra to solve for x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our answer by replacing 9 for x in the original equation.
256=49−5=44
Continue searching for examples and problems online, and if you use the laws of exponents, you will turn into a master of these concepts, working out most exponential equations with no issue at all.
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