Question

Solve 4^(x+2) = 12 for x using the change of base formula log base b of y equals log y over log b.

Answer 1

To solve the equation

`4^(x+2) = 12`

for x, you can use the change of base formula log base b of y equals log y over log b. By following the steps, you should find the solution to be x ≈ -0.086.

Step-by-step explanation:

To solve the equation

`4^(x+2) = 12`

for x, we can use the change of base formula log base b of y equals log y over log b. Let's solve it step by step:

1. Take the logarithm of both sides of the equation using any base (let's use base 10 for this example):
2. 
   `log (4^(x+2))`
3. = log 12
4. Apply the power rule of logarithms: (x+2) * log 4 = log 12
5. Divide both sides by log 4: x+2 = log 12 / log 4
6. Subtract 2 from both sides to isolate x: x = log 12 / log 4 - 2
7. Use a calculator to evaluate the right side of the equation: x ≈ -0.086

Therefore, the solution to the equation

`4^(x+2) = 12`

is approximately x = -0.086.

Answer 2

`x \approx -0.208`

Step-by-step explanation:

By applying logarithms to each side of the equation and some algebraic handling:

`x+2 = \log_(4) 12`

`x + 2 = (\log_(10) 12)/(\log_(10) 4)`

`x = (\log_(10)12)/(\log_(10)4) - 2`

`x \approx -0.208`