Exponential EquationsDefinition, Solving, and Examples
In math, an exponential equation occurs when the variable shows up in the exponential function. This can be a terrifying topic for kids, but with a some of instruction and practice, exponential equations can be worked out easily.
This blog post will talk about the explanation of exponential equations, types of exponential equations, proceduce to work out exponential equations, and examples with solutions. Let's get started!
What Is an Exponential Equation?
The primary step to solving an exponential equation is determining 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 major things to look for when you seek to establish 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, look at this equation:
y = 3x2 + 7
The primary thing you must observe is that the variable, x, is in an exponent. The second thing you should notice is that there is another term, 3x2, that has the variable in it – just not in an exponent. This means that this equation is NOT exponential.
On the contrary, take a look at this equation:
y = 2x + 5
Yet again, the primary thing you must notice is that the variable, x, is an exponent. Thereafter thing you must observe is that there are no other value that have the variable in them. This means that this equation IS exponential.
You will come across exponential equations when you try solving diverse calculations in compound interest, algebra, exponential growth or decay, and other functions.
Exponential equations are crucial in arithmetic and play a central duty in figuring out many math questions. Therefore, it is crucial to completely grasp what exponential equations are and how they can be used as you move ahead in mathematics.
Kinds of Exponential Equations
Variables come in the exponent of an exponential equation. Exponential equations are amazingly ordinary in daily life. There are three major kinds of exponential equations that we can work out:
1) Equations with the same bases on both sides. This is the most convenient to work out, as we can easily set the two equations same as each other and figure out for the unknown variable.
2) Equations with different bases on both sides, but they can be made similar utilizing properties of the exponents. We will show some examples below, but by converting the bases the same, you can observe the described steps as the first case.
3) Equations with distinct bases on both sides that cannot be made the same. These are the toughest to work out, but it’s feasible utilizing the property of the product rule. By raising two or more factors to the same power, we can multiply the factors on both side and raise them.
Once we have done this, we can resolute the two latest equations identical to one another and figure out the unknown variable. This article does not contain logarithm solutions, but we will let you know where to get assistance at the very last of this blog.
How to Solve Exponential Equations
Knowing the definition and kinds of exponential equations, we can now learn to solve any equation by ensuing these easy steps.
Steps for Solving Exponential Equations
Remember these three steps that we are going 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 are in common base. Then, we can solve them using standard algebraic methods.
Lastly, we have to solve for the unknown variable. Since we have solved for the variable, we can put this value back into our original equation to figure out the value of the other.
Examples of How to Solve Exponential Equations
Let's check out a few examples to note how these process work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can notice that all the bases are identical. Hence, all you need to do is to restate the exponents and solve utilizing algebra:
y+1=3y
y=½
So, we change the value of y in the given equation to support that the form is real:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a more complex question. 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 working includes breaking down respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to come to the ultimate answer:
28=22x-10
Perform algebra to work out the x in the exponents as we performed in the prior example.
8=2x-10
x=9
We can double-check our workings by replacing 9 for x in the initial equation.
256=49−5=44
Keep looking for examples and questions on the internet, and if you utilize the rules of exponents, you will turn into a master of these concepts, figuring out almost all exponential equations with no issue at all.
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Working on problems with exponential equations can be tough without help. While this guide goes through the fundamentals, you still may encounter questions or word problems that may hinder you. Or perhaps you desire some extra help as logarithms come into play.
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