91Ó°ÊÓ

The exponent rule is a fundamental principle when dealing with logarithms. It provides a way to express roots using exponents. In essence, the nth root of a number can be written as an exponent by raising it to the power of its reciprocal. For example, for the fifth root, which is denoted as \( \sqrt[5]{a} \), it can be expressed as \( a^{1/5} \).
This rule allows us to convert roots into powers, making it possible to use logarithmic properties effectively.
Thus, an expression like \( \log_2 \sqrt[5]{\frac{xy^4}{16}} \) is first transformed into \( \log_2 \left(\frac{xy^4}{16}\right)^{1/5} \).

• Remember that roots are just another way to express powers.
• This transformation simplifies the use of other logarithmic rules.

The power rule of logarithms is an incredibly useful tool. It states that if you have a logarithm of a number raised to a power, you can bring that power in front as a multiplication factor. This is mathematically expressed as:\[ \log_a(n^p) = p \log_a n \] This rule allows us to simplify expressions significantly by pulling powers out of logarithms.
In our example, \( \log_2 ((\frac{xy^4}{16})^{1/5}) \) can be rewritten as \( \frac{1}{5} \log_2 (\frac{xy^4}{16}) \).

• This rule turns complex exponential expressions into simpler ones by using multiplication.
• It's especially helpful for solving and simplifying logarithmic expressions.

The quotient rule is key when you have a logarithm of a fraction or division. When you encounter such a case, the rule helps you separate the fraction into two logarithmic terms. The rule is described by:\[\log_b(\frac{m}{n}) = \log_b m - \log_b n\] This means you can split the logarithm of a division into two logs subtracted from each other.
In our problem, \( \frac{1}{5} \log_2 (\frac{xy^4}{16}) \) becomes \( \frac{1}{5} (\log_2 (xy^4) - \log_2 16) \).

• Use the quotient rule to split fractions under a logarithm.
• Breaking the expression into two parts makes it easier to apply further rules.

The product rule is a vital tool for simplifying expressions inside a logarithm, especially with multiplication. According to this rule, the logarithm of a product can be split into a sum of logarithms. Mathematically, it is written as:\[\log_b(mn) = \log_b m + \log_b n\] So, if you have a product inside your logarithm, you can separate each factor into a separate term. In the example we're discussing, \( \log_2 (xy^4) \) is split into \( \log_2 x + \log_2 y^4 \).
Furthermore, we simplify \( \log_2 16 = 4 \) since 16 is a power of 2. We further use the power rule to simplify \( \log_2 y^4 \) into \( 4\log_2 y \).
Hence, the expression is fully expanded to \( \frac{1}{5} (\log_2 x + 4\log_2 y - 4) \).

• Apply the product rule when you have factors within a logarithm.
• This rule helps break down complex expressions step by step.