# Chapter 9 - Section 9.4 - Properties of Logarithms - Exercise Set - Page 713: 52

$\ln{\frac{x^7}{y^3}}$

#### Work Step by Step

The product rule for logarithms says that $log_b{MN}=log_bM+log_bN$ i.e. the logarithm of a product is the sum of the logarithms. The quotient rule for logarithms says that $log_b{\frac{M}{N}}=log_bM-log_bN$ i.e. the logarithm of a quotient is the difference of the logarithms. The power rule for logarithms says that $\log_b{M^p}=p\log_bM$ i.e. the logarithm of a number with an exponent is the exponent times the logarithm of the number. $\log_ba=\frac{\log_ca}{\log_cb}$ Hence here: $7\ln x-3\ln y=\ln{x^7}-\ln{y^3}=\ln{\frac{x^7}{y^3}}$

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