Problem 77 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\left(\frac{2 \log _{b} x}{3}+\frac{3 \log _{b} y}{4}\right)-5 \log _{b} z\)

Step-by-Step Solution

Answer

The single logarithm of the expression \(\left(\frac{2 \log _{b} x}{3}+\frac{3 \log _{b} y}{4}\right)-5 \log _{b} z\) is \(\log _{b} \left(\frac{x^{\frac{2}{3}}y^{\frac{3}{4}}}{z^{5}}\right)\).

Step 1: Apply the Power Rule

The power rule is used first. According to this rule, the coefficients in front of the logarithms should be moved to the exponent position inside the logarithms. Therefore, \(\left(\frac{2 \log _{b} x}{3}+\frac{3 \log _{b} y}{4}\right)-5 \log _{b} z\) is rewritten into \(\log _{b} x^{\frac{2}{3}}+\log _{b} y^{\frac{3}{4}}-\log _{b} z^{5}\).

Step 2: Apply the Quotient Rule

The quotient rule states that the subtraction of logarithms is equivalent to a division inside one logarithm. According to this rule, the expression \(\log _{b} x^{\frac{2}{3}}+\log _{b} y^{\frac{3}{4}}-\log _{b} z^{5}\) is changed into \(\log _{b} \left(\frac{x^{\frac{2}{3}} * y^{\frac{3}{4}}}{z^{5}}\right)\).

Step 3: Combine the Logarithms

As the last step, the two logarithms \(\log _{b} x^{\frac{2}{3}}\) and \(\log _{b} y^{\frac{3}{4}}\) are combined into a single logarithm as a product, according to the product rule. Now the final expression is \(\log _{b} \left(\frac{x^{\frac{2}{3}}y^{\frac{3}{4}}}{z^{5}}\right)\).

Key Concepts

Power Rule in Logarithms

The power rule in logarithms is a handy tool for simplifying expressions. Specifically, it allows us to transform a logarithmic term by moving the coefficient in front of a logarithm to the exponent of the term inside the logarithm. This transformation can be represented as follows: when you have an expression like \( c \log_{b} a \), it can be rewritten as \( \log_{b} a^{c} \).
For example, if you encounter \( \frac{2}{3} \log_{b} x \), using the power rule, it becomes \( \log_{b} x^{\frac{2}{3}} \). Moving the coefficient to the exponent significantly simplifies the expression, making it easier to manipulate in complex logarithmic operations.
This rule is not only a simplification tool but also essential when expressing multiple logarithmic terms into a single one. It sets the stage for applying further rules, like the quotient and product rules, necessary for writing complex logarithmic expressions as a singular log function.

Quotient Rule in Logarithms

The quotient rule in logarithms is essential when dealing with the subtraction of logarithmic expressions. It allows the subtraction of two logarithms to be expressed as the log of a quotient. In other words, from two logs being subtracted, \( \log_{b} a - \log_{b} c \), it becomes \( \log_{b} \left( \frac{a}{c} \right) \).
This rule is powerful as it transforms complex differences into a single, more manageable logarithmic term. Let's consider an example, using the previously simplified expressions: When you have \( \log_{b} x^{\frac{2}{3}} - \log_{b} z^{5} \), the quotient rule helps to convert this into a single log, \( \log_{b} \left( \frac{x^{\frac{2}{3}}}{z^{5}} \right) \).
By applying the quotient rule, calculations and manipulations become simpler, enabling easier solving of logarithmic equations and contributing to overall efficiency in handling logs.

Product Rule in Logarithms

The product rule for logarithms is quite useful for adding two logarithmic expressions into one. This means when two logarithms with the same base are added, say \( \log_{b} a + \log_{b} c \), they can be combined using the product rule to become \( \log_{b} (a \cdot c) \).
Following the application of the power and quotient rules, the product rule allows us to manage the addition of logs in an expression. For instance, in our original problem: After applying the power and quotient rules, if you have \( \log_{b} x^{\frac{2}{3}} + \log_{b} y^{\frac{3}{4}} \), the product rule combines these into \( \log_{b} (x^{\frac{2}{3}} \cdot y^{\frac{3}{4}}) \).
Using the product rule efficiently combines multiple logarithmic elements into one, resulting in a streamlined single log expression. This is especially beneficial when trying to condense and simplify equations during the problem-solving process.