Problem 55 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 5 \ln x-2 \ln y $$

Step-by-Step Solution

Answer

The simplified expression is \(\ln \frac{x^5}{y^2}\).

Step 1: Apply the Logarithm's Power Rule

Use power rule of logarithms to move the coefficients in front of the logs to become the power of their respective variable. This changes the expression to \(\ln x^5 - \ln y^2\).

Step 2: Apply the Logarithm's Quotient Rule

The quotient rule says that the log of a division is the difference of the logs. Thus, we can combine the two logarithms into a single log, giving the final simplified version of the expression as \(\ln \frac{x^5}{y^2}\).

Key Concepts

Logarithm Power Rule

The logarithm power rule is an essential tool in simplifying logarithmic expressions. This rule allows us to transform products in front of a logarithm into exponents of the logarithmic argument. It is a straightforward but powerful concept.

When you have a logarithmic expression such as \( c \cdot \log_b(a) \), the power rule shows that it is equivalent to \( \log_b(a^c) \).
Put simply, multiply the coefficient by the logarithm and convert it to an exponent.

• Example: If you have \( 5 \ln x \), it can be rewritten as \( \ln(x^5) \).
• This makes the job of simplifying much easier, as moving a coefficient to an exponent can often lead to further simplifications.

Understanding this rule helps break down larger expressions into more manageable parts, especially when dealing with multiple logarithms.

Logarithm Quotient Rule

The logarithm quotient rule is another useful technique for simplifying expressions that involve subtracting two logarithms. This rule states that the difference between two logarithms can be rewritten as a single logarithm of a quotient.

In mathematical terms, if you have \( \log_b(m) - \log_b(n) \), it can be expressed as \( \log_b\left(\frac{m}{n}\right) \).

• This conversion is especially handy, as it consolidates expressions into a single logarithm.
• For example, returning to our expression \( \ln x^5 - \ln y^2 \), applying the quotient rule results in \( \ln \left(\frac{x^5}{y^2}\right) \).

This not only simplifies the expression but also makes it easier to evaluate if necessary.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions is all about using the properties of logarithms, like the power and quotient rules, to make the expressions more compact and manageable. The goal is usually to rewrite the expression in a way that uses the fewest number of logs possible.

To simplify logarithmic expressions:

• Apply the power rule to move coefficients as exponents.
• Use the quotient rule to combine logs into a single expression where appropriate.
• Evaluate the expression if possible, or leave it in a simplified form.

In our example, by applying these techniques, we condensed \( 5 \ln x - 2 \ln y \) into a single logarithmic form: \( \ln \left(\frac{x^5}{y^2}\right) \).
The process of simplifying not only reduces the complexity but also makes further computations less cumbersome, particularly when substituting numerical values in place of variables.