Question

(Remember to show work and explain.)

Find the inverse of the functions.

$y = ln(x - 1)$
$y = ln \: 2x$
$y = {e}^{3x}$
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Answer 1

1)

↪y=ln(x-1)

Interchange role of x and y,

↪x=ln (y-1)

Changing it to exponential form,

↪e^x=y-1

↪y=e^x+1

↪inverse of f(x)=y=e^x +1

2)

↪y=ln(2x)

Interchange role of x and y,

↪x=ln(2y)

Changing it to exponential form,

↪e^x=2y

↪y=(e^x)/2=inverse of f(x)

3)

↪y=e^3x

Interchange role of x and y,

↪x=e^3y

Take log on both sides,

↪ln x= ln {(e)^3y}

I have used property of log here to further simplify...

↪ln x= 3y × ln(e)

↪ln x=3y × 1 {as ln e=1}

↪y=ln(x)/3 =inverse of f(x)