Question

What is the solution to the equation 9^(x+1) =27

Answer 1

`x = (1)/(2)`

Step-by-step explanation

The given exponential equation is

`{9}^(x + 1) = 27`

The greatest common factor of 9 and 27 is 3.

We rewrite the each side of the equation to base 3.

`{3}^(2(x + 1)) = {3}^(3)`

Since the bases are equal, we can equate the exponents.

`2(x + 1) = 3`

Expand the parenthesis to get:

`2x + 2 = 3`

Group similar terms

`2x = 3 - 2`

`2x = 1`

`x = (1)/(2)`

Answer 2

For this case we must solve the following equation:

`9 ^ {x + 1} = 27`

We rewrite:

`9 = 3 * 3 = 3 ^ 2\\27 = 3 * 3 * 3 = 3 ^ 3`

Then the expression is:

`3^ {2 (x + 1)} = 3 ^ 3`

Since the bases are the same, the two expressions are only equal if the exponents are also equal. So, we have:

`2 (x + 1) = 3`

We apply distributive property to the terms within parentheses:

`2x + 2 = 3`

Subtracting 2 on both sides of the equation:

`2x = 3-2\\2x = 1`

Dividing between 2 on both sides of the equation:

`x = \frac {1} {2}`

Answer:

`x = \frac {1} {2}`