Partial Fraction Question: Simplifying ln(2) and ln(3) to ln(8/3) in 8/5 form

[stripedcat]

Without being too concerned how we got there. The answer to a partial fraction question a friend and I are doing is

24/5 ln(2) - 8/5ln(3)

The system does not accept this answer however, it wants the simplified form

8/5 ln (8/3)

We're not sure how to get that form.

More specifically

8/5 (ln(8)-ln(3))

It's the ln(8) we're not sure about.

 

[MarkFL]

If you and your friend's answer was:

\(\displaystyle \frac{24}{5}\ln(2)-\frac{8}{5}\ln(3)\)

then it would be equivalent to the answer expected by "the system."

Recall the logarithmic property:

\(\displaystyle b\cdot\log_{a}(c)=\log_{a}\left(c^b\right)\)

 

[Euge]

If you had $(24/5)\ln{2} - (8/5)\ln{3}$ instead of $(26/5)\ln{2} - (8/5)\ln{3}$ you would get the desired $(8/5)\ln{8/3}$. To see this, use the formula $b \ln{a} = \ln{a^b}$ to write

$(24/5)\ln{2} = (8/5)(3\ln{2}) = (8/5) \ln{2^3} = (8/5)\ln{8}$.

Then $(24/5)\ln{2} - (8/5)\ln{3} = (8/5)(\ln{8} - \ln{3})$, which, as you mentioned, equals $(8/5)\ln{8/3}$.

 

[stripedcat]

Sooo

24/5 ln(2) - 8/5 ln(3)

(24/5)/(8/5)=3

8/5 (3ln(2) - ln(3))

2^3 = 8

ln(8)-ln(3)

8/5 ln(8/3)

Ace.