Questions: Express as a sum or difference of logarithms without exponents. logb (x^3 / (y^5 z^6))^(1/5)

Solution Steps

To express the given logarithmic expression as a sum or difference of logarithms without exponents, we can use the properties of logarithms. Specifically, we will use the power rule, the quotient rule, and the product rule of logarithms.

1. Apply the power rule to bring the exponent outside the logarithm.
2. Apply the quotient rule to separate the logarithm of the numerator and the denominator.
3. Apply the product rule to further separate the logarithms of the individual terms in the denominator.

We start with the expression \( \log_{b} \sqrt[5]{\frac{x^{3}}{y^{5} z^{6}}} \). Using the power rule of logarithms, we can rewrite this as: \[ \log_{b} \left( \frac{x^{3}}{y^{5} z^{6}} \right)^{\frac{1}{5}} = \frac{1}{5} \log_{b} \left( \frac{x^{3}}{y^{5} z^{6}} \right) \]

Next, we apply the quotient rule, which states that \( \log_{b} \left( \frac{A}{B} \right) = \log_{b}(A) - \log_{b}(B) \). Thus, we have: \[ \frac{1}{5} \left( \log_{b}(x^{3}) - \log_{b}(y^{5} z^{6}) \right) \]

Now, we apply the product rule to the term \( \log_{b}(y^{5} z^{6}) \): \[ \log_{b}(y^{5} z^{6}) = \log_{b}(y^{5}) + \log_{b}(z^{6}) \] This gives us: \[ \frac{1}{5} \left( \log_{b}(x^{3}) - \left( \log_{b}(y^{5}) + \log_{b}(z^{6}) \right) \right) \]

We can further simplify each logarithmic term using the power rule: \[ \log_{b}(x^{3}) = 3 \log_{b}(x), \quad \log_{b}(y^{5}) = 5 \log_{b}(y), \quad \log_{b}(z^{6}) = 6 \log_{b}(z) \] Substituting these back into our expression, we get: \[ \frac{1}{5} \left( 3 \log_{b}(x) - (5 \log_{b}(y) + 6 \log_{b}(z)) \right) \]

This simplifies to: \[ \frac{1}{5} \left( 3 \log_{b}(x) - 5 \log_{b}(y) - 6 \log_{b}(z) \right) = \frac{3}{5} \log_{b}(x) - \log_{b}(y) - \frac{6}{5} \log_{b}(z) \]

Final Answer