91Ó°ÊÓ

The logarithm of a quotient simplifies to a difference of two logarithms. This is an important property in logarithmic calculations and is expressed as: \[\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\]This rule is very useful when dealing with complex fractions inside logarithmic expressions. Let's consider the original example from the exercise: \[\log_3\left(\frac{\sqrt[3]{y}}{7 x}\right)\]By applying the quotient rule, we break it down into two logarithmic parts:

• \(\log_3(\sqrt[3]{y})\)
• \(- \log_3(7x)\)

Each part is now simpler to handle individually, which eventually makes it easier to solve or simplify further expressions. This fundamental property helps in converting multiplicative or divisional logarithmic forms into additive or subtractive ones.

When it comes to the logarithm of a power, you can use the exponent property to simplify expressions. This property states:\[\log_b(M^n) = n\cdot \log_b(M)\]This relationship allows you to take the exponent and multiply it by the logarithm of the base value. For our exercise, we applied this rule to the numerator:\[\log_3(\sqrt[3]{y}) \]Since \(\sqrt[3]{y}\) can also be written as \(y^{1/3}\), we use the power rule:

• \(\log_3(\sqrt[3]{y}) = \log_3(y^{1/3}) = \frac{1}{3}\log_3(y)\)

This effectively simplifies our logarithm, revealing the hidden power relations within the expression. By doing this, you can take complex powers and express them more straightforwardly, providing deeper insights into logarithmic transformations.

The logarithm of a product can be broken down into the sum of individual logarithms. This transformation is simplified as:\[\log_b(MN) = \log_b(M) + \log_b(N)\]Applying this rule helps manage expressions where multiple variables or constants multiply within the logarithmic function. In the exercise, we see its use on:\[\log_3(7x)\]By breaking the expression into separate logarithms, we obtain:

• \(\log_3(7x) = \log_3(7) + \log_3(x)\)

This separation into additive components streamlines further mathematical treatment and manipulation of complex logarithmic expressions. Breaking them apart in this way illuminates the multiplicative relationships underlying the expression, which is a powerful tool for solving or transforming logarithmic equations.