Problem 46 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to combine the expression. $$ 2\left(\log _{5} x+2 \log _{5} y-3 \log _{5} z\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_5\left(\frac{x^2 \cdot y^4}{z^6}\right) \).

Step 1: Apply the Power Rule

The Power Rule of logarithms states that \( a \cdot \log_b(m) = \log_b(m^a) \). We first apply this to each term inside the parentheses: \( 2(\log_5 x) = \log_5(x^2) \), \( 2(2\log_5 y) = \log_5(y^4) \), and \( 2(-3\log_5 z) = \log_5(z^{-6}) \).

Step 2: Combine Using the Product Rule

The Product Rule states that \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \). Using this rule, we combine \( \log_5(x^2) + \log_5(y^4) \) into \( \log_5(x^2 \cdot y^4) \).

Step 3: Combine Using the Quotient Rule

The Quotient Rule states that \( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n}) \). Now, apply this to include the last term: \( \log_5(x^2 \cdot y^4) - \log_5(z^6) = \log_5\left(\frac{x^2 \cdot y^4}{z^6}\right) \).

Step 4: Final Expression

Now that all terms are combined, the final expression, \( 2(\log_5 x + 2\log_5 y - 3\log_5 z) \), simplifies to \( \log_5\left(\frac{x^2 \cdot y^4}{z^6}\right) \).

Key Concepts

Power Rule

The Power Rule of logarithms is a powerful tool for simplifying expressions. It states that if you have a multiplication factor in front of a logarithm, you can rewrite it as an exponent inside the log. The rule is expressed as: \(a \cdot \log_b(m) = \log_b(m^a)\). This means you can move the coefficient in front of the log as an exponent on the argument of the log.

Let's see how this works with an example. Suppose you have the expression \(2 \cdot \log_5(x)\). By applying the Power Rule, this becomes \(\log_5(x^2)\). Similarly, for \(2 \cdot 2 \cdot \log_5(y)\), it becomes \(\log_5(y^4)\). This conversion is particularly useful for simplifying complex logarithmic expressions as it changes them into one compact form where all exponents are inside the log.

Keep in mind:

• The base of the logarithm remains unchanged while applying the Power Rule.
• This rule only applies to multiplication factors (coefficients) and not to terms being added or subtracted.

Product Rule

The Product Rule comes into play when you have addition within logarithms. It helps in consolidating multiple log terms into a single expression. The Product Rule can be stated as: \(\log_b(m) + \log_b(n) = \log_b(m \cdot n)\). Hence, when you add logs of the same base, it is equivalent to the log of the product of their arguments.

Imagine you have \(\log_5(x^2) + \log_5(y^4)\). By applying the Product Rule, you can combine these into \(\log_5(x^2 \cdot y^4)\). With this rule, you're turning addition into a multiplication inside a single logarithm, simplifying the expression significantly.

To effectively use the Product Rule:

• Ensure all logarithms have the same base before applying the rule.
• Always double-check that terms being combined are indeed logs.

Quotient Rule

The Quotient Rule is one of the key strategies for managing subtraction in logarithms. It allows you to express the difference of two log terms as the log of a quotient. The formula for this rule is: \(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\). This means when you subtract one log from another (with the same base), you can rewrite it as the log of the division of their arguments.

Consider the expression \(\log_5(x^2 \cdot y^4) - \log_5(z^6)\). By employing the Quotient Rule, this becomes \(\log_5\left(\frac{x^2 \cdot y^4}{z^6}\right)\). This method effectively transforms subtraction into a division within one log expression.

Helpful tips for the Quotient Rule:

• Again, make sure the bases of the logs match before applying the rule.
• Carefully handle negative signs; they imply division when applying this rule.