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

Question

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.

Answer 1

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

Answer 2

Hints:

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

Answer 3

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.

Answer 4

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

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, $\ln x+ \ln y=\ln {xy}$

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

Final Answer,

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