Problem 39 - Precalculus

Question

Write each expression as a single logarithm. $$5 \log _{b} u-2 \log _{b} v$$

Step-by-Step Solution

Answer

The expression is \( \log_b\left(\frac{u^5}{v^2}\right) \).

Step 1: Apply the Power Rule for Logarithms

The power rule for logarithms states that for any logarithm \( a \cdot \log_b(x) = \log_b(x^a) \). For the term \( 5 \log_b(u) \), apply the power rule to rewrite it as \( \log_b(u^5) \). Similarly, for \( 2 \log_b(v) \), rewrite it as \( \log_b(v^2) \).

Step 2: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). Apply this rule to the transformed expression \( \log_b(u^5) - \log_b(v^2) \), giving \( \log_b\left(\frac{u^5}{v^2}\right) \).

Step 3: Combine into a Single Logarithm

The expression is now expressed as a single logarithmic term: \( \log_b\left(\frac{u^5}{v^2}\right) \). This is the solution that combines the original expression into one logarithm.

Key Concepts

Power Rule

The power rule for logarithms is a fundamental principle that helps simplify logarithmic expressions when you have a coefficient in front of a logarithm. The rule states that a logarithm with a product in front of it, like \( a \cdot \log_b(x) \), can be rewritten as \( \log_b(x^a) \). This means you move the coefficient as an exponent of the argument of the logarithm.
Let's apply this to a simple example. Suppose you have \( 3 \log_2(7) \). According to the power rule, this expression can be rewritten as \( \log_2(7^3) \). Simplifying further, \( 7^3 = 343 \), so the expression becomes \( \log_2(343) \).
- **Use the power rule** whenever you see a multiplication in front of a logarithm.- **Simplifying expressions** this way often makes them easier to combine or manipulate later on.Using the power rule also sets the stage for combining multiple logarithms into a single expression, which is quite handy in solving complex logarithmic problems.

Quotient Rule

The quotient rule for logarithms is another fundamental tool for simplifying expressions. It allows you to combine the subtraction of two logarithms into a single logarithm. The rule can be expressed as \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). This means when you subtract one logarithm from another, the result is a logarithm of a quotient, i.e., one item divided by the other.
**Example Application:** Imagine you have two logarithms, \( \log_3(27) - \log_3(9) \). By the quotient rule, this expression becomes \( \log_3\left(\frac{27}{9}\right) \). Once you simplify the division inside the logarithm, \( \frac{27}{9} = 3 \), so the expression simplifies to \( \log_3(3) \).

• The quotient rule is helpful for simplifying expressions and solving equations involving logarithms.
• Use the quotient rule when you see logarithms being subtracted with the same base.

The quotient rule thus helps to express complex logarithmic expressions in a more manageable form by reducing them into single logarithmic terms.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms, which are the inverse operations of exponentiation. They are useful for simplifying expressions, solving equations, and performing calculations in various fields such as science, engineering, and finance.
A **logarithmic expression** might involve addition, subtraction, or multiplication of logarithms. To simplify these expressions, you can use rules such as the power rule and quotient rule described above. These tools allow you to combine or break down logarithmic expressions into simpler forms. For instance, when you start with the expression \( 5 \log_b(u) - 2 \log_b(v) \), applying both the power rule and the quotient rule simplifies it to a single logarithm: \( \log_b\left(\frac{u^5}{v^2}\right) \).

• Always ensure you are working with logarithms of the same base when applying these rules.
• Remember the properties of exponents and how they relate to logarithms, as they often simplify your work.
• Practicing these rules on different expressions will make it easier to spot patterns and apply the correct simplifications quickly.

Logarithmic expressions are everywhere, and understanding how to manipulate them is a fundamental skill in mathematics.