Problem 82 - Algebra and Trigonometry

Question

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: \(3 \log _{a} x+2 \log _{a} y-5 \log _{a} z\)

Step-by-Step Solution

Answer

\ \ \ \log_a\left(\frac{x^3 \cdot y^2}{z^5}\right) .

Step 1: Apply the Power Rule

Use the power rule of logarithms that states: ewline ewline \( c \log_b(m) = \log_b(m^c) \ \) ewline ewline This transforms the expression into ewline ewline \ \ \ \ \log_a(x^3) + \log_a(y^2) - \log_a(z^5) .

Step 2: Combine Using the Addition Rule

Utilize the addition rule of logarithms: ewline ewline \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \ \) ewline ewline This gives us: ewline ewline \ \ \ \ \ \log_a(x^3 \cdot y^2) - \log_a(z^5) .

Step 3: Combine Using the Subtraction Rule

Now apply the subtraction rule of logarithms: ewline ewline \( \log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right) \ \) ewline ewline This results in: ewline ewline \ \ \ \ \ \ \ \log_a\left(\frac{x^3 \cdot y^2}{z^5}\right) .

Key Concepts

Power Rule

When working with logarithmic expressions, the power rule is extremely useful. It allows you to move a coefficient in front of a logarithm to become an exponent inside the logarithm. This can simplify the expression significantly. The rule states:

\( c \, \log_b(m) = \log_b(m^c) \)

So, for our specific problem, you can transform each term by moving the coefficients inside the logarithms as exponents. This changes:

• \(3 \, \log_a(x)\) into \(\log_a(x^3)\)
• \(2 \, \log_a(y)\) into \(\log_a(y^2)\)
• \( -5 \, \log_a(z)\) into \(\log_a(z^5)\)

Now, the expression becomes \(\log_a(x^3) + \log_a(y^2) - \log_a(z^5)\).

Addition Rule

The addition rule in logarithms is another handy tool for simplifying expressions. When you add two logarithms with the same base, you can combine them into a single logarithm by multiplying their arguments. The rule is:

\( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)

Applying this rule to our transformed expression, \( \log_a(x^3) + \log_a(y^2)\), leads to:
\log_a(x^3 \cdot y^2)\

This simplifies the addition part of the expression. Now we just have one term left to deal with.

Subtraction Rule

The subtraction rule for logarithms lets you combine two logarithms into one by dividing their arguments, as long as they have the same base. The rule is:

\( \log_b(m) - \log_b(n) = \log_b(\frac{m}{n})\ \)

In our case, we use this rule on the simplified addition: \log_a(x^3 \cdot y^2) - \log_a(z^5)\

According to the subtraction rule:
\log_a(\frac{x^3 \cdot y^2}{z^5})\

This final expression succinctly combines all initial logarithmic terms into a single logarithm: \log_a(\frac{x^3 \cdot y^2}{z^5})\.