# Thread: Need help with Logarithm and Inverse Function

1. ## Need help with Logarithm and Inverse Function

I am having difficulties understanding how to solve the following problems:

1) Express as a single logarithm and, if possible, simplify:

$\displaystyle \frac{3}{2}ln4x^6 - \frac{4}{5}ln2y^{10}$

The book shows the answer as: $\displaystyle ln\frac{2^{11/5}x^9}{y^8}$

I understand how the $\displaystyle x^6$ turns into $\displaystyle x^9$, but how does the $\displaystyle 4$ in $\displaystyle \frac{3}{2}ln4x^6$ turn into $\displaystyle 2^{11/5}$?

2) Express as a single logarithm:

$\displaystyle 7lnx + 3(lny^2 - lnz^3)$

Since #1 is similar to #2, I am unsure on how to correctly solve this problem.

3) Find the inverse function of:

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

When I do the math, it shows that the inverse function is exactly the same as the original function? How is that possible?

I would appreciate it greatly if you could answer the questions above step-by-step.

Thanks

2. Originally Posted by Ineedhelpwithlogs
I am having difficulties understanding how to solve the following problems:

1) Express as a single logarithm and, if possible, simplify:

$\displaystyle \frac{3}{2}ln4x^6 - \frac{4}{5}ln2y^{10}$

The book shows the answer as: $\displaystyle ln\frac{2^{11/5}x^9}{y^8}$

I understand how the $\displaystyle x^6$ turns into $\displaystyle x^9$, but how does the $\displaystyle 4$ in $\displaystyle \frac{3}{2}ln4x^6$ turn into $\displaystyle 2^{11/5}$?
$\displaystyle \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})$

$\displaystyle \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}$

$\displaystyle \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f rac{4}{5}}}\right)$

Now simplify

3. Originally Posted by Ineedhelpwithlogs
3) Find the inverse function of:

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

When I do the math, it shows that the inverse function is exactly the same as the original function? How is that possible?

I would appreciate it greatly if you could answer the questions above step-by-step.

Thanks
$\displaystyle y = f(x) = \frac{3x}{x-3}$

swap x and y and then solve for y

$\displaystyle x = \frac{3y}{y-3}$

$\displaystyle (y-3)x = 3y$

$\displaystyle yx-3x = 3y$

$\displaystyle -3x = 3y-yx$

$\displaystyle -3x = y(3-x)$

Can you solve for y?

4. Originally Posted by pickslides
$\displaystyle \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})$

$\displaystyle \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}$

$\displaystyle \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f rac{4}{5}}}\right)$

Now simplify
Simplifying is where I get lost - sorry, I should have mentioned that.

5. Originally Posted by pickslides
$\displaystyle y = f(x) = \frac{3x}{x-3}$

swap x and y and then solve for y

$\displaystyle x = \frac{3y}{y-3}$

$\displaystyle (y-3)x = 3y$

$\displaystyle yx-3x = 3y$

$\displaystyle -3x = 3y-yx$

$\displaystyle -3x = y(3-x)$

Can you solve for y?
$\displaystyle y = \frac{-3x}{3 - x}$
$\displaystyle f^{-1}(x) = \frac{-3x}{3 - x}$

Thank you, pickslides. I see where I made my mistake.

6. $\displaystyle \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})$

$\displaystyle \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}$

$\displaystyle \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f rac{4}{5}}}\right)$

$\displaystyle \ln\left(\frac{4^{\frac{3}{2}}x^{6\times \frac{3}{2}}}{2^{\frac{4}{5}}y^{10\times\frac{4}{5 }}}\right)$

7. Originally Posted by pickslides
$\displaystyle \frac{3}{2}\ln(4x^6) - \frac{4}{5}\ln(2y^{10})$

$\displaystyle \ln(4x^6)^{\frac{3}{2}} - \ln(2y^{10})^{\frac{4}{5}}$

$\displaystyle \ln\left(\frac{(4x^6)^{\frac{3}{2}}}{(2y^{10})^{\f rac{4}{5}}}\right)$

$\displaystyle \ln\left(\frac{4^{\frac{3}{2}}x^{6\times \frac{3}{2}}}{2^{\frac{4}{5}}y^{10\times\frac{4}{5 }}}\right)$

$\displaystyle \ln\left(\frac{4^{\frac{3}{2}}x^{9}}{2^{\frac{4}{5 }}y^{8}}\right)$

then divide 4 by 2 to get:

$\displaystyle \ln\left(\frac{2^{\frac{3}{2}}x^{9}}{{\frac{4}{5}} y^{8}}\right)$

Now, I don't know what to do next (and I'm not sure if that last step was correct).