An equation is** exponential** if the variable (unknown) appears in the exponent of a power. To solve this type of equations we have to remember all the potentiation properties. In some cases variable changes are made to make your solution easier. The general expression of a ** exponential equation** is the next:

Where, a and b are real numbers, x is the exponent of the expression and the unknown of the equation.

Knowing that the general expression of an exponential equation is **a ^{x} - b = 0**We will look for a way to find the value of the unknown variable through the application of clearances and power properties.

## How to solve an exponential equation?

To understand how to solve such an equation, we must perform some exercises:

### Exponential Equation Exercises

**Exercise 1**: Find the value of x in the following equation 2^{x + 1} = 8.

We will first transform 8 into base 2 power:

We rewrite the equation:

Since the bases are equal, the exponents are equalized and we solve for x:

Now, let's check that this value of x is a solution to the equation:

So x = 2 is the solution to our exponential equation.

**Exercise 2**: Find the value of x in the following equation 3^{5x + 2} = 6561.

We transform 6561 into base 3 power:

We rewrite the equation:

Since the bases are equal, the exponents are equalized and we solve for x:

To verify that x = 6/5 is a solution, we substitute it in the equation:

So x = 6/5 is the solution to our exponential equation.

**Exercise 3**: Find the value of x in the following equation 5^{x2 + 5x }= 1/625.

We will first express 625 in base 5 power:

We rewrite the equation

Applying the property to^{-n} = 1 / a^{n}, we have:

Since the bases are equal, the exponents are equalized:

Remaining an equation of second degree:

Using the quadratic formula: We will have the values of x solution to the exponential equation:

**Exercise 4**: Find the value of x in the following equation:

Applying the power property to^{n}·to^{m} = to^{n + m} we have:

We take common factor 2^{x }:

To terms 2^{- two} and 2^{- 3} we apply the power property to^{-n} = 1 / a^{n}:

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