# Thread: Find the equation of the inverse function..Logarithm.

1. ## Find the equation of the inverse function..Logarithm.

1a)
$\displaystyle y=log_5 x$
$\displaystyle x=log_5 y$
$\displaystyle 5^x=y$
$\displaystyle y=5^x$
$\displaystyle f^{-1} (x)=5^x$

b)
$\displaystyle y=-log_5 (-x)$
$\displaystyle x=-log_5 (-y)$
$\displaystyle -5^x=-y$
$\displaystyle -y=-5^x$
$\displaystyle f^{-1} (-x)=-5^x$

Do those seem correct?

2. Originally Posted by NotSoBasic
1a)
$\displaystyle y=log_5 x$
$\displaystyle x=log_5 y$
$\displaystyle 5^x=y$
$\displaystyle y=5^x$
$\displaystyle f^{-1} (x)=5^x$

b)
$\displaystyle y=-log_5 (-x)$
$\displaystyle x=-log_5 (-y)$
$\displaystyle -5^x=-y$
$\displaystyle -y=-5^x$
$\displaystyle f^{-1} (-x)=-5^x$

Do those seem correct?

I agree with the first question.

I think the 2nd question should be
$\displaystyle y=-log_5 (-x)$
$\displaystyle x=-log_5 (-y)$
$\displaystyle -x=log_5 (-y)$
$\displaystyle 5^{-x}=-y$
$\displaystyle -y=5^{-x}$
$\displaystyle y=-5^{-x}$
$\displaystyle f^{-1}=-5^{-x}$

3. Hello, NotSoBasic!

The first one is correct . . .

$\displaystyle b) \;\;y \:=\:-\log_5 (-x)$

We have: .$\displaystyle x \:=\:-\log_5(-y)$

Switch sides: .$\displaystyle -\log_5(-y) \:=\:x$

Multiply by -1: .$\displaystyle \log_5(-y) \:=\:-x$

Exponentiate: .$\displaystyle -y \:=\:5^{-x}$

Multiply by -1: .$\displaystyle y \:=\:-5^{-x}$

Therefore: .$\displaystyle f^{-1}(x) \:=\:-5^{-x}$

Edit: too slow again . . .
.

4. Those make more sense, thanks guys!

5. Originally Posted by Soroban

Edit: too slow again . . .

happens often to me as well.