What is the inverse of the function f(x)=2^x+6

1) The inverse of the function f(x)=2^x+6 is: f^(-1) (x)=log(x-6) / log(2)

2) The inverse of the functio f(x)=2^(x+6) is: f^(-1) (x) =log(x) / log(2) - 6

Solution:

1) f(x)=2^x+6

y=f(x)

y=2^x+6

Solving for x: Subtracting 6 both sides of the equation:

y-6=2^x+6-6

y-6=2^x

Applying log both sides of the equation:

log(y-6)=log(2^x)

Applying poperty of logarithm: log(a^b)=b log(a); with a=2 and b=x

log(y-6)=x log(2)

Dividing both sides of the equation by log(2)

log(y-6) / log(2)=x log(2) / log(2)

log(y-6) / log(2)=x

x=log(y-6) / log(2)

Changing "x" by "f^(-1) (x)" and "y" by "x":

f^(-1) (x)=log(x-6) / log (2)

2) f(x)=2^(x+6)

y=f(x)

y=2^(x+6)

Solving for x: Applying log both sides of the equation:

log(y)=log(2^(x+6))

Applying poperty of logarithm: log(a^b)=b log(a); with a=2 and b=x+6

log(y)=(x+6) log(2)

Dividing both sides of the equation by log(2)

log(y) / log(2)=(x+6) log(2) / log(2)

log(y) / log(2)=x+6

Subtracting 6 both sides of the equation:

log(y) / log(2) - 6 = x+6-6

log(y) / log(2) - 6 = x

x=log(y) / log(2) -6

Changing "x" by "f^(-1) (x)" and "y" by "x":

f^(-1) (x)=log(x) / log (2) - 6