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.
Hint: $r\ln x=\ln(x^r)$, so $4\ln(x)=\ln(x^4)$. Also, $\ln(x)+\ln(y)+\ln(z)=\ln(xyz)$.
$4\ln(x)+2\ln(x^4y^3)+5\ln(z)$

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, 

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