Problem 85 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ 2 \log _{b} x+4 \log _{b} y-3 \log _{b} z $$

Step-by-Step Solution

Answer

\( \log_b \left( \frac{x^2 y^4}{z^3} \right) \)

Step 1: Apply the Power Rule

To simplify each term, apply the power rule of logarithms: \( k \log_b a = \log_b a^k \). This gives us:\[ 2 \log_b x = \log_b x^2 \]\[ 4 \log_b y = \log_b y^4 \]\[ -3 \log_b z = \log_b z^{-3} \]

Step 2: Combine Logarithms

Now that we have expressed each term using the power rule, we can combine them into a single logarithm using the product and quotient rules.The combined expression becomes:\[ \log_b x^2 + \log_b y^4 - \log_b z^3 = \log_b\left(\frac{x^2y^4}{z^3}\right) \]

Step 3: Write the Final Expression

The expression can now be written as a single logarithm:\[ \log_b \left( \frac{x^2 y^4}{z^3} \right) \] This is the simplified expression of the given logarithmic terms.

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is an essential tool used to simplify expressions that involve logarithms. It states that for any real number \( k \) and positive number \( a \), \( k \log_b a = \log_b a^k \). This powerful rule helps us transform the logarithmic expression by moving the constant factor in front of the logarithm into an exponent inside the log.

For example, using the power rule, the expression \( 3 \log_b x \) can be rewritten as \( \log_b x^3 \). Similarly, \( -3 \log_b z \) becomes \( \log_b z^{-3} \). This transformation makes it much easier to use other logarithmic rules to combine and simplify complex expressions. By mastering the power rule, you can make seemingly complicated problems much simpler to handle.

Product Rule of Logarithms

The product rule of logarithms allows us to combine two logarithms with the same base when adding them. According to this rule: if \( \log_b A \) and \( \log_b B \) are two logarithms with the same base, then \( \log_b A + \log_b B = \log_b (A \cdot B) \). This rule leverages the property that multiplying inside the logarithm corresponds to addition outside of it.

When given a problem that asks you to simplify a sum of logarithms, apply this rule to combine the terms. For instance, with expressions like \( \log_b x^2 + \log_b y^4 \), we use the product rule to combine them into a single log: \( \log_b (x^2 \cdot y^4) \). This step moves us closer to creating a single, simplified expression by combining factors inside the logarithm itself.

Quotient Rule of Logarithms

The quotient rule of logarithms is useful when dealing with the subtraction of two logarithms. This rule indicates that the subtraction of two same-base logarithms can be expressed as the division of their arguments inside a single logarithm: \( \log_b A - \log_b B = \log_b \left(\frac{A}{B}\right) \).

This is especially helpful when simplifying complex expressions. For instance, when you need to simplify \( \log_b x^2 + \log_b y^4 - \log_b z^3 \), the quotient rule helps by turning the subtraction into a division within a single log: \( \log_b \left(\frac{x^2 y^4}{z^3}\right) \). This rule is particularly powerful in ensuring that we can express multiple terms as a singular, cohesive expression, easy to comprehend and analyze.