1. ## Logarithms as Inverses.

Please solve with work and explain why and what you are doing

Write the Inverse of each function:

1) Y= log3X

2) Y= 2^(x/3)

2. Originally Posted by north1224
Please solve with work and explain why and what you are doing

Write the Inverse of each function:

1) Y= log3X

2) Y= 2^(x/3)

note that $\displaystyle \log_a b = c \Longleftrightarrow a^c = b$

3. Originally Posted by north1224
Please solve with work and explain why and what you are doing

Write the Inverse of each function:

1) Y= log3X

2) Y= 2^(x/3)

$\displaystyle y=log_{3}(x)$

exchange the x and y and then solve for y

$\displaystyle x=log_{3}(y) \iff 3^x=y$

$\displaystyle f^{-1}(x)=3^x$

$\displaystyle y=2^{x/3}$ swapping the var

$\displaystyle x=2^{y/3} \iff log_{2}(x)=\frac{y}{3} \iff 3log_{2}(x)=y$

$\displaystyle f^{-1}(x)=3log_2(x)$

4. $\displaystyle y=2^{x/3}$ swapping the var

$\displaystyle x=2^{y/3} \iff log_{2}(x)=\frac{y}{3} \iff 3log_{2}(x)=y$
Where did the 2 come from?

5. Originally Posted by north1224
Where did the 2 come from?
the 2 was in the original question. it ended up as the base of the logarithm based on the rule i gave you earlier

6. Originally Posted by Jhevon
the 2 was in the original question. it ended up as the base of the logarithm based on the rule i gave you earlier
I Meant the two from "TheEmptySet" description, Thanks anyway, The rule helped me alot for the rest of my worksheet though, thanks for that

7. It is the rule that Jhevon mentioned above.

8. Originally Posted by TheEmptySet
It is the rule that Jhevon mentioned above.
No, TheEmptySet, Where did the 2 as a variable come from.?

9. Originally Posted by north1224
No, TheEmptySet, Where did the 2 as a variable come from.?
I don't understand 2 is a constant not a variable?

$\displaystyle y=2^{\frac{x}{3}}$

The 2 is the base of eponential function.

if we interchange the variables we get

$\displaystyle x=2^{\frac{y}{3}}$ rewritting as a log we get

$\displaystyle log_{2}(x)=\frac{y}{3} \iff y= 3 log_{2}(x)$

Originally Posted by TheEmptySet
I don't understand 2 is a constant not a variable?

$\displaystyle y=2^{\frac{x}{3}}$
$\displaystyle x=2^{\frac{y}{3}}$ rewritting as a log we get
$\displaystyle log_{2}(x)=\frac{y}{3} \iff y= 3 log_{2}(x)$