Answer in Algebra for Vidhi #109493

Consider the function “y=3^{x+6}”

Interchange the variable “x” and “y” to obtain,

“x=3^{y+6}”

Take log of base “e” both sides as,

“ln( x)=ln( 3^{y+6})”

Use the power property of logarithm “ln( a^{m})=mln( a)” to simplify the right hand side of the equation as,

“ln( x)=(y+6 )ln( 3)”

“ln( x)=yln( 3)+6ln( 3)”

“ln( x)-6ln( 3)=yln( 3)”

“y=frac{ln( x)-6ln( 3)}{ln( 3)}”

The sketch of the curve and its inverse is as shown in the figure below:

Here, the curve and it’s inverse is symmetric about the line “y=x” .

Therefore, the inverse of the function “y=3^{x+6}” is “y=frac{ln( x)-6ln( 3)}{ln( 3)}”