Questions: Express in terms of logarithms without exponents. [ log bleft(sqrt[4]fracm^20 n^20b^5right) ]

Solution Steps

To express the given logarithmic expression without exponents, we can use the properties of logarithms and exponents. First, simplify the expression inside the logarithm by applying the rules of exponents. Then, use the logarithm power rule to bring the exponents outside the logarithm.

The original expression is \(\log_{b}\left(\sqrt[4]{\frac{m^{20} n^{20}}{b^{5}}}\right)\). We start by simplifying the expression inside the logarithm. The fourth root can be expressed as a power of \(\frac{1}{4}\):

\[ \left(\frac{m^{20} n^{20}}{b^{5}}\right)^{\frac{1}{4}} \]

Apply the power rule \((a^m)^n = a^{m \cdot n}\) to simplify the expression:

\[ \left(\frac{m^{20} n^{20}}{b^{5}}\right)^{\frac{1}{4}} = \frac{m^{5} n^{5}}{b^{\frac{5}{4}}} \]

Now, apply the properties of logarithms. The logarithm of a quotient is the difference of the logarithms:

\[ \log_{b}\left(\frac{m^{5} n^{5}}{b^{\frac{5}{4}}}\right) = \log_{b}(m^{5} n^{5}) - \log_{b}(b^{\frac{5}{4}}) \]

Use the power rule of logarithms, \(\log_{b}(a^n) = n \cdot \log_{b}(a)\):

\[ = 5\log_{b}(m) + 5\log_{b}(n) - \frac{5}{4}\log_{b}(b) \]

Since \(\log_{b}(b) = 1\), the expression simplifies to:

\[ = 5\log_{b}(m) + 5\log_{b}(n) - \frac{5}{4} \]

Final Answer