Exponential equation

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:

\ large \ mathbf {a ^ {x} -b = 0}

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

exponential equation

Knowing that the general expression of an exponential equation is ax - b = 0We 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 2x + 1 = 8.

We will first transform 8 into base 2 power:

\ large 8 = 2 \ cdot 2 \ cdot 2 = 2 ^ {3}

We rewrite the equation:

\ large 2 ^ {+ 1} = 2 <3>

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

\ large x + 1 = 3

\ large x = 3-1

\ large x = 2

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

\ large 2 2 + 1 = 2 3

\ large 2 ^ {3} = 2 ^ {3}

\ large 8 = 8

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

Exercise 2: Find the value of x in the following equation 35x + 2 = 6561.

We transform 6561 into base 3 power:

\ large 6561 = 3 \ cdot 3 \ cdot 3 \ cdot 3 \ cdot 3 \ cdot 3 \ cdot 3 \ cdot 3 \ cdot 3 = 3 ^ {8}

We rewrite the equation:

\ large 3 5x + 2 = 3 8

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

\ large 5x + 8 = 2

\ large 5x = 8-2

\ large x = \ frac {6} {5}

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

\ large 3 ^ {5 \ left (\ frac {6} {5} \ right) +2} = 6561

\ large 3 8 = 6561

\ large 6561 = 6561

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

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

We will first express 625 in base 5 power:

\ large 625 = 5 \ cdot 5 \ cdot 5 \ cdot 5 = 5 ^ {4}

We rewrite the equation\ large 5 ^ {x ^ {2} + 5x} = \ frac {1} {5 ^ {4}}

Applying the property to-n = 1 / an, we have:

\ large 5 x 2 + 5x = 5 -4

Since the bases are equal, the exponents are equalized:

\ large x ^ 2 -5x = -4

Remaining an equation of second degree:

\ large x ^ 2 -5x + 4 = 0

Using the quadratic formula:\ large x = -b \ pm \ sqrt {\ frac {b ^ {2} -4ac} {2a}} We will have the values of x solution to the exponential equation:

\ large x_ {1} = 1

\ large x_ {2} = 4

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

\ large 2 x + 2 x + 1 + 2 x-2 + 2 x-3 = 864

Applying the power property ton·tom = ton + m  we have:

\ large 2 x {2} {2} 2 + 2 x {2} - 2 + 2 x {x} - 3 = 864

We take common factor 2x :

\ large 2 ^ {x} \ cdot \ left (1 + 2 + 2 ^ {- 2} +2 ^ {- 3} \ right) = 864

To terms 2- two and 2- 3  we apply the power property to-n = 1 / an:

\ large 2 ^ {x} \ cdot \ left (1 + 2 + \ frac {1} {2 ^ {2}} + \ frac {1} {2 ^ {3}} \ right) = 864

\ large 2 ^ {x} \ cdot \ left (1 + 2 + \ frac {1} {4} + \ frac {1} {8} \ right) = 864

\ large 2 ^ {x} \ cdot \ left (\ frac {27} {8} \ right) = 864

\ large 2 ^ {x} = \ cdot \ left (\ frac {864 \ cdot 8} {27} \ right)

\ large 2 ^ {x} = 256

\ large 2 ^ {x} = 2 ^ {8}

\ large x = 8

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