Problem 70 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write as a single logarithm: \(\log _{7} x+3 \log _{7} y-\log _{7}(x+y)\)

Step-by-Step Solution

Answer

\( \log_{7}(\frac{x \cdot y^3}{x + y}) \)

Step 1 - Use the power rule for logarithms

Apply the power rule of logarithms: The power rule states that 3 \log_{7}(y) = \log_{7}(y^{3}). So rewrite the given expression: \( \log_{7}(x) + 3 \log_{7}(y) - \log_{7}(x + y) = \log_{7}(x) + \log_{7}(y^3) - \log_{7}(x + y) \)

Step 2 - Use the product rule for logarithms

Apply the product rule of logarithms: The product rule states that \(\log_{b}(a) + \log_{b}(c) = \log_{b}(a \cdot c)\). So combine \log_{7}(x) and \log_{7}(y^3): \( \log_{7}(x) + \log_{7}(y^3) = \log_{7}(x \cdot y^3) \)

Step 3 - Use the quotient rule for logarithms

Apply the quotient rule of logarithms: The quotient rule states that \(\log_{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c})\). So combine \log_{7}(x \cdot y^3) and \log_{7}(x + y): \( \log_{7}(x \cdot y^3) - \log_{7}(x + y) = \log_{7}(\frac{x \cdot y^3}{x + y}) \)

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a fundamental rule that helps make complex logarithmic expressions simpler. When you have a logarithm with an exponent, you can move the exponent in front of the logarithm. This rule can be expressed as: \ \( \log_{b}(a^{c}) = c \log_{b}(a)\ \).
In our exercise, we used this rule to rewrite the term \ \( 3 \log_{7}(y) \). Applying the power rule, we get: \ \( 3 \log_{7}(y) = \log_{7}(y^{3}) \).
This transformation simplifies handling logarithmic expressions by reducing the number of terms in the equation.

Product Rule of Logarithms

The product rule of logarithms is useful when you need to combine the logarithms of multiple numbers. According to this rule: \ \( \log_{b}(a) + \log_{b}(c) = \log_{b}(a \cdot c)\ \).
This allows you to merge two separate logarithms into a single logarithm of the product of their arguments.
In the step-by-step solution, we applied this rule to \ \( \log_{7}(x) + \log_{7}(y^{3}) = \log_{7}(x \cdoty^{3}) \).
By using this rule, we combined the logarithms into a single term, making our expression more manageable.

Quotient Rule of Logarithms

The quotient rule of logarithms is handy for combining logarithms that involve division. This rule states: \ \( \log_{b}(a) - \log_{b}(c) = \log_{b}( \frac{a}{c} )\ \). Combining logarithms this way can simplify expressions and make them more straightforward to work with.
In our exercise, we used the quotient rule on \ \( \log_{7}(x \cdot y^{3}) - \log_{7}(x + y)\ \). Applying the quotient rule, we get: \ \( \log_{7}( \frac{x \cdot y^{3}}{x + y} )\ \).
This final step combined our terms into a single logarithmic expression, which is much easier to interpret and use for further calculations.