Question 149932
*[Tex \LARGE 4\left\[\ln(z)+\ln(z+5)\right\]-2\ln(z-5)] Start with the given expression.



*[Tex \LARGE 4\ln(z)+4\ln(z+5)-2\ln(z-5)] Distribute.



*[Tex \LARGE \ln(z^4)+\ln\left((z+5)^4\right)-\ln\left((z-5)^2\right)] Rewrite each term using the identity  {{{ln(x^y)=y*ln(x))}}}



*[Tex \LARGE \ln\left(z^4(z+5)^4\right)-\ln\left((z-5)^2\right)] Combine the logs using the identity {{{ln(A)+ln(B)=ln(A*B)}}}



*[Tex \LARGE \ln\left(\frac{z^4(z+5)^4}{(z-5)^2}\right)] Combine the logs using the identity {{{ln(A)-ln(B)=ln(A/B)}}}



So *[Tex \LARGE 4\left\[\ln(z)+\ln(z+5)\right\]-2\ln(z-5)] simplifies to *[Tex \LARGE \ln\left(\frac{z^4(z+5)^4}{(z-5)^2}\right)]