Problem 87 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \frac{1}{2} \log _{b} x-\log _{b} x+4 \log _{b} y $$

Step-by-Step Solution

Answer

The expression is \( \log_b\left(\frac{y^4}{\sqrt{x}}\right) \).

Step 1: Apply the Power Rule

The power rule of logarithms states that \( a \log_b(x) = \log_b(x^a) \). First, apply this rule to each term: \( \frac{1}{2} \log_b(x) \) becomes \( \log_b(x^{1/2}) \), and \( 4 \log_b(y) \) becomes \( \log_b(y^4) \).

Step 2: Simplify Using the Power Rule Results

Now, rewrite the expression using the results of the power rule: \( \log_b(x^{1/2}) - \log_b(x) + \log_b(y^4) \).

Step 3: Apply the Quotient Rule

The quotient rule of logarithms states that \( \log_b(a) - \log_b(b) = \log_b(\frac{a}{b}) \). Apply this to the first two terms: \( \log_b(x^{1/2}) - \log_b(x) = \log_b(\frac{x^{1/2}}{x}) \).

Step 4: Simplify the Fraction

Simplify the fraction: \( \frac{x^{1/2}}{x} = x^{-1/2} \). Thus, the expression becomes \( \log_b(x^{-1/2}) + \log_b(y^4) \).

Step 5: Apply the Product Rule

The product rule of logarithms states that \( \log_b(a) + \log_b(b) = \log_b(ab) \). Use this on the simplified terms: \( \log_b(x^{-1/2}) + \log_b(y^4) = \log_b(x^{-1/2}y^4) \).

Step 6: Final Expression

Combine all parts to get the final expression: \( \log_b(\frac{y^4}{\sqrt{x}}) \).

Key Concepts

Power Rule of Logarithms

Logarithms are incredibly useful for simplifying expressions, and one of the most powerful tools in this process is the power rule. The power rule states that if you have a logarithm multiplied by a number, you can rewrite this as an exponent inside the logarithm. In other words, for any positive real number \( a \), \( a \log_b(x) = \log_b(x^a) \). This concept allows us to take coefficients of logarithmic terms and transform them into exponents, simplifying expressions.

• Example: \( 3 \log_b(x) \) can be rewritten as \(\log_b(x^3) \).
• This is particularly useful when dealing with expressions involving multiple logarithmic terms, as it enables us to condense them into a single, more manageable form.

By applying the power rule, we can efficiently handle complex logarithmic expressions and help make further simplifications possible, such as using the quotient or product rules.

Quotient Rule of Logarithms

Once you've rewritten terms using the power rule, the next step is often to apply the quotient rule. This rule allows you to simplify the difference of two logarithms into a single logarithm involving a fraction. Specifically, the quotient rule asserts that \( \log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right) \).

• Example: Transform \( \log_b(x) - \log_b(y) \) into \( \log_b\left(\frac{x}{y}\right) \).
• The quotient rule is particularly useful when the initial expression involves subtraction, enabling combining of terms.

By using the quotient rule on expressions where we have negative terms, we can simplify them into more concise forms, setting the stage for eventual full simplification using other rules like the product rule.

Product Rule of Logarithms

After using the power and quotient rules, you'll often find expressions that still need further simplification. The product rule helps us here by combining separately added logarithms into a single logarithmic expression. According to the product rule, \( \log_b(a) + \log_b(b) = \log_b(ab) \).

• Example: Combine \(\log_b(x) + \log_b(y)\) into \( \log_b(xy) \).
• This rule is particularly helpful when the logistic expressions include addition operations, allowing for the merging of multiple terms.

Using the product rule, you further reduce complex logarithmic expressions into a single, simple log statement. Together with the power and quotient rules, the product rule provides the tools needed to express even complicated logarithmic problems in a clearer, more concise manner.