91Ó°ÊÓ

The logarithm power rule is a handy tool when dealing with logarithmic expressions. It states that if you have a constant multiplier in front of a log, you can simplify the expression using exponents. Specifically, the rule is expressed as \( c \log_b M = \log_b M^c \). This means that instead of having the constant \( c \) in front of the log, you raise the argument \( M \) to the power of \( c \) inside the log.

Let's break down how this works using the expression from our exercise: \( 5 \log_a x \) can be rewritten as \( \log_a (x^5) \). Here, the constant 5 becomes an exponent of \( x \).

Similarly, for \(-\frac{1}{2} \log_a (3x-4)\), using the logarithm power rule, we convert it to \( \log_a ((3x-4)^{-1/2}) \). The negative sign in front of \( \frac{1}{2} \) contributes to the negative exponent inside the log.

Combining logarithms is another essential skill, especially when you need to express multiple logs as a single logarithmic term. The main rules for combining logarithms include:

• Addition of logs: \( \log_b M + \log_b N = \log_b (M \cdot N) \)
• Subtraction of logs: \( \log_b M - \log_b N = \log_b \left( \frac{M}{N} \right) \)

When applying these rules, it’s crucial to keep operations correct.

In the exercise, after applying the power rule, you have transformed expressions like \( \log_a (x^5) - \log_a ((3x-4)^{-1/2}) - \log_a ((5x+1)^{-3}) \). Combining these using the subtraction rule gives you a single log:\[ \log_a \left( \frac{x^5}{(3x-4)^{-1/2} \cdot (5x+1)^{-3}} \right) \].

Understanding the properties of exponents is pivotal when simplifying the expression inside a logarithm. Some key exponent rules include:

• \( a^{-b} = \frac{1}{a^b} \) — This means that a negative exponent indicates reciprocal.
• Multiplying powers: \( x^a \cdot x^b = x^{a+b} \) — Combine like bases by adding exponents.

In our exercise, the expression can be further simplified using these rules.

The log expression \( \log_a \left( \frac{x^5}{(3x-4)^{-1/2} \cdot (5x+1)^{-3}} \right) \) simplifies to \( \log_a ( x^5 \cdot (3x-4)^{1/2} \cdot (5x+1)^3 ) \) because dividing by \((3x-4)^{-1/2} \) and \((5x+1)^{-3} \) is the same as multiplying by \((3x-4)^{1/2} \) and \((5x+1)^{3} \), respectively.

Understanding these exponent properties allows you to execute steps in complex logarithmic simplification effortlessly.