Express $4\ln(x)+2\ln(x^4y^3)+5\ln(z)$ as a single logarithm

The problem is to express $4\ln(x)+2\ln(x^4y^3)+5\ln(z)$ as a single logarithm.

Our teacher has shown us examples for the same base and when it's both add and subtract. But I'm not sure how to do this.

• Can you elaborate on your title? Does the final result need to be in log base-3? Commented Apr 29, 2014 at 0:51
• I think Blake just wants to: "express the sum of three logarithms as a single logarithm," but the title is a little unclear. Commented Apr 29, 2014 at 0:54
• Sorry, I was only wanting to express this as a single logarithm. :) Commented Apr 29, 2014 at 0:57
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– user122519
Commented Apr 29, 2014 at 4:34

Hint: $r\ln x=\ln(x^r)$, so $4\ln(x)=\ln(x^4)$. Also, $\ln(x)+\ln(y)+\ln(z)=\ln(xyz)$.

• Good answer (+1), but it's unclear from the OP whether he needs his final answer in $\log_3$. Commented Apr 29, 2014 at 0:50
• @KajHansen Hmm, yeah, that's true. I read the title and interpreted it as log base $3$. If not, then the last sentence is redundant.
– user122283
Commented Apr 29, 2014 at 0:53
• Sorry, I do not need my answer in log3. I've gotten to ln(x^4)+ln(x^4y^3)^2+ln(z)^5. Commented Apr 29, 2014 at 0:53
– user122283
Commented Apr 29, 2014 at 0:54
• I'm not 100% confident in this answer, but I got ln(x^16y^3z^5)^10. The ^10 at the end is making me unsure of the process I'm using. Commented Apr 29, 2014 at 0:56

Hints:

$\log(x) + \log(y) = \log(xy)$ and $n\log(x) = log(x^n)$ for logarithms of any base.

Are you familiar with the identity $r \ln x = \ln (x^r)$?

I don't believe you need to change the base of any logarithms here, by the way.

• Oops, I think I might've misinterpreted his question. I thought the final result had to be in $\log_3$ due to the title. Commented Apr 29, 2014 at 0:47

$4\ln(x)+2\ln(x^4y^3)+5\ln(z)$

By using identity, $a\times ln x=ln x^a$

$\implies \ln(x)^4+\ln{(x^4y^3)}^2+\ln(z)^5$

By using identity, $\ln x+ \ln y=\ln {xy}$

$\implies \ln(x)^{4+8}y^{3\times2}(z)^5$

$\implies \ln(x)^{12}y^{6}(z)^5$