Final answer to the problem
$\ln\left(x\right)+\frac{1}{2}\ln\left(1+5x\right)+\ln\left(2\right)+\ln\left(3\right)-11\ln\left(x-3\right)$
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Step-by-step Solution
Learn how to solve expanding logarithms problems step by step online. Expand the logarithmic expression ln((6x(1+5x)^(1/2))/((x-3)^11)). The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x). Applying the product rule for logarithms: \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right). Applying the product rule for logarithms: \log_b\left(MN\right)=\log_b\left(M\right)+\log_b\left(N\right).
Final answer to the problem
$\ln\left(x\right)+\frac{1}{2}\ln\left(1+5x\right)+\ln\left(2\right)+\ln\left(3\right)-11\ln\left(x-3\right)$
