Problem 27 - Algebra 2 and Trigonometry

Question

Write each expression as a single logarithm. \(\log _{10} y-2 \log _{10}(y-1)\)

Step-by-Step Solution

Answer

\( \log_{10} \left( \frac{y}{(y-1)^2} \right) \)

Step 1: Apply Logarithm Power Rule

The expression given is \( \log_{10} y - 2 \log_{10} (y-1) \). The logarithm power rule states that \( a \log_b x = \log_b (x^a) \). Therefore, we can rewrite the second term as \( \log_{10} ((y-1)^2) \). This simplifies the expression to \( \log_{10} y - \log_{10} ((y-1)^2) \).

Step 2: Use Logarithm Quotient Rule

The logarithm quotient rule tells us that \( \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right) \). Applying this to our expression \( \log_{10} y - \log_{10} ((y-1)^2) \), we get \( \log_{10} \left( \frac{y}{(y-1)^2} \right) \).

Key Concepts

Logarithm Power Rule

Understanding the logarithm power rule can simplify many logarithmic expressions, and it's especially useful when dealing with exponential factors. The rule states that for any logarithm \( a \log_b x \), you can convert the expression to \( \log_b (x^a) \). Here, the coefficient becomes the exponent.
📌 **For example:**

• Given an expression like \( 2 \log_{10}(y-1) \), apply the power rule to rewrite it as \( \log_{10}((y-1)^2) \).

This technique reveals the hidden power within the expression without altering the base logarithm. This step is pivotal when aiming to express a log statement more compactly.

Logarithm Quotient Rule

After applying the power rule, you might find yourself with expressions that subtract one logarithm from another. This is where the logarithm quotient rule comes in handy.
The quotient rule states that \( \log_b a - \log_b c = \log_b \left(\frac{a}{c}\right) \). This effectively converts a subtraction into the division inside a logarithm.
📌 **For example:**

• Consider the expression \( \log_{10} y - \log_{10} ((y-1)^2) \). By the quotient rule, it transforms into \( \log_{10} \left( \frac{y}{(y-1)^2} \right) \).

This technique merges the logs into one, making the expression simpler and easy to interpret.

Simplifying Logarithmic Expressions

Combining logarithm rules helps to condense expressions into simpler forms. Practicing this process increases familiarity with these transformations and aids in solving complex problems.
To simplify a logarithmic expression:

• Apply the power rule first to express any coefficients as exponents.
• Next, use the quotient rule or other relevant rules to combine terms.

This approach reduces complexity and helps you to understand the underlying relationships in logarithmic expressions.
The expression \( \log_{10} y - 2 \log_{10}(y-1) \) beautifully illustrates this method by transforming into \( \log_{10} \left( \frac{y}{(y-1)^2} \right) \). Mastering these rules equips you with the ability to tackle any logarithmic challenge efficiently.