The name itself suggests that the exponent equation must involve exponents. It is so obvious that the number of times a base number is multiplied is indicated by its exponent. But sometimes the power number is a variable. Wherever power is in variable form and it is part of an equation, we state it as an exponential equation. To solve such equations, mathematicians use the connection between the exponents and the logarithms. Below in this blog, we will tell you **how to solve exponent equations**. But before that, let us look at what the definition of an exponent equation has to say.

**WHAT ARE EXPONENTIAL EQUATIONS?**

An exponential equation is one in which the exponent, or a portion of the exponent, is a variable. If we have to state a few examples, 5^{x}= 125, 7^{x-3}= 745, 4^{8y-2}= 448, and many more are all exponential equations. There are three main types of exponential equations. The three types are

i) The exponential equations with the same base on both sides. An example of such equations would be 5^{6}=5^{y}.

ii) The exponential equations that can be modified to have the same base on both sides. For example, 3^{x}= 27 can be written as 3^{x}= 3^{3}.

iii) The exponential equations that can not be modified to have the same base on both sides. For example, 6^{x}= 245.

**HOW TO SOLVE EXPONENT EQUATIONS?**

EXPONENTIAL EQUATION FORMULA.

There can be two cases, one in which both bases are the same and another in which the bases are not the same. In either of these cases, we have a formula that is helpful in solving the exponential equations. We have explained it below in detail:

The Exponential Equations Formula says:

i) When bases are the same, we can apply:

a^{x} = a^{y} <==> x = y

ii) When bases are not the same, we can apply:

b^{x} = a <==> log_{b}a = x

Apply logarithm on both sides.

**EXPONENTIAL EQUATIONS TO LOGARITHMIC FORM.**

Exponents are logarithms and vice versa. Hence a logarithmic equation of an exponential equation may be transformed. This helps to solve an exponential equation using several bases. The formula is used for converting exponential equations to logarithmic.

b^{x }= a <==> log_{b}a =x

**HOW TO SOLVE EXPONENT EQUATION WITH THE SAME BASE.**

If the case is, wherein, an exponential equation both the sides of the equation have the same base. For instance, in an exponent equation 6^{x} = 6^{5}, we have the same base of 6 on both sides. Many times exponents on both are not the same but we can easily modify them to be the same. For instance, 8^{x} = 16, in this equation bases are not the same but we can write it like 8^{x} = 8^{2} (because 8^{2} = 16) so that we have the same base. In each of these situations, we apply only the property of equality of exponential equations to resolve the exponential equations. We solve for the variable by setting the exponents to the same value.

SOLVING EXPONENTIAL EQUATIONS WITH THE SAME BASES.

a^{x} = a^{y }<==> x = y

For Example,

3^{x} = 3^{6 }<==> x = 6

Let’s take another example, where the base is again not the same but can be modified.

- Let us try solving this exponential equation 8
^{x+1}= 512^{x}.

We know that 512 = 8^{3}. Keeping this in mind, the given equation can be written as.

8^{x+1} = (8^{3})^{x}

8^{x+1} = 7^{3x}

Now that bases on both sides are the same. So we can set the exponents to be the same.

X + 1 = 3x

Subtracting x from both sides,

2x = 1

Dividing both sides by 2,

x = ½.

**HOW TO SOLVE EXPONENT EQUATION WITH DIFFERENT BASE.**

In some cases, wherein the exponential equations do not have bases on both sides but can be modified to make them the same. In such a case, we use the logarithm method to solve exponential equations. For instance, 8^{x} = 3 both of the sides have the same base on both sides nor they can be modified. If it is the case, we can do the mentioned below things:

- Use the formula to transform the exponential equation into the logarithmic form.

b^{x} = a <==>, a = x

Using this formula, you can solve the given exponent equation.

- To both ends of the equation, apply logarithm (log) and solve for the variable. In such a case, we have a formula known as property of logarithm, log a
^{m}= m log a.

Let us solve this equation, 6^{x} = 4, using both of these methods.

METHOD 1: Let us first convert 6^{x} = 4 in logarithm form.

So, we get, log_{6}4 = x

Using the change of base property,

x = (log 4)/ (log 6)

METHOD 2: We will apply the log on both sides 6^{x} = 4.

Then we get log 6^{x} = log 4

Using the property log a^{m} = m log an on the side of the equation, we get

X log 6 = log 4

Dividing both sides by log 6,

x = (log 4)/ (log 6)

**ADDITIONAL TIPS ON HOW TO SOLVE EXPONENT EQUATIONS.**

- The same bases can only be set equal to the exponential equations.
- Apply logarithm on both sides to solve the exponential equations of various bases.
- Equations with the same foundations can also be solved by logarithms.
- If there’s 1 on any side of an exponential equation, we may write 1 = a
^{0}for any ‘a.’

To solve 5x = 1, for instance, we may write 5x = 5^{0}, and x = 0.

- We may either use “log” or use “ln” on either side to resolve the exponential equation by utilizing logarithms.

**FREQUENTLY ASKED QUESTIONS (FAQs)**

1. What is an exponent equation?

Ans. An exponential equation is one in which the exponent, or a portion of the exponent, is a variable. If we have to state a few examples are 5^{x}= 125, 7^{x-3}= 745, 4^{8y-2}= 448, and many more are all exponential equations. There are three main types of exponential equations.

2. How to solve exponent equations?

Ans. You can use either of the two formulas according to the case.

The Exponential Equations Formula says:

i) When bases are the same, we can apply:

a^{x} = a^{y} <==> x = y

ii) When bases are not the same, we can apply:

b^{x} = a <==> log_{b}a = x

Apply logarithm on both sides.

3. What are the three types of exponent equations?

Ans.

i) The exponential equations with the same base on both sides.

ii) The exponential equations that can be modified to have the same base on both sides.

iii) The exponential equations that can not be modified to have the same base on both sides.