Problem 70 - Algebra 2

Question

Write each expression as a single logarithm. \(\log _{5} x-\frac{1}{5} \log _{5} y\)

Step-by-Step Solution

Answer

The expression \(\log _{5} x-\frac{1}{5} \log _{5} y\) as a single logarithm is \(\log _{5} \(\frac{x}{ y^\frac{1}{5}}\)\).

Step 1: Analyze the given expression

The given expression is \(\log _{5} x-\frac{1}{5} \log _{5} y\). It contains two logarithms with the same base (5) that are subtracted from each other.

Step 2: Apply power rule of logarithms

Apply the power rule of logarithms to the second part of the expression. This means moving the coefficient \(\frac{1}{5}\) inside the logarithm and raising the argument y to that power. By doing so, \(\frac{1}{5} \log _{5} y\) becomes \(\log _{5} y^\frac{1}{5}\). The expression then becomes \(\log _{5} x - \log _{5} y^\frac{1}{5}\).

Step 3: Apply subtraction rule of logarithms

Apply the rule of subtraction for logarithms to the expression. This states that the rule turns a subtraction of logarithms into a division inside one logarithm. By applying that rule, \(\log _{5} x - \log _{5} y^\frac{1}{5}\) becomes \(\log _{5} \(\frac{x}{ y^\frac{1}{5}}\)\). This is the final single logarithm expression.

Key Concepts

Power Rule of Logarithms

In the world of logarithms, there's a very handy tool known as the power rule. This rule tells us how to deal with coefficients in front of log expressions. If you see an expression like \(c \log_b a\), you can use the power rule to rewrite it as \( \log_b a^c \). This means you move the coefficient \(c\) up as an exponent in the argument of the log. In our example, we had \(\frac{1}{5} \log_5 y\). Using the power rule, we raised \(y\) to the power of \(\frac{1}{5}\), changing the expression to \(\log_5 y^{\frac{1}{5}}\). This makes it easier to combine log terms as we'll see in the next sections. By understanding and using the power rule, you can simplify and manipulate logarithmic expressions effectively.

Subtraction Rule of Logarithms

One useful property of logarithms is how subtraction between two logs can be simplified. The subtraction rule is our guide here. This says that \( \log_b a - \log_b c\) can be rewritten as \( \log_b \frac{a}{c} \).This rule is particularly useful when dealing with expressions that involve subtraction. In the original exercise, after applying the power rule, the expression was \( \log_5 x - \log_5 y^{\frac{1}{5}} \). The subtraction rule allowed us to combine these into a single logarithmic expression: \( \log_5 \left( \frac{x}{y^{\frac{1}{5}}} \right) \). Understanding this rule not only helps with simplifying expressions, but it also paves the way for more advanced algebraic manipulations.

Simplifying Logarithms

Simplifying logarithmic expressions means using rules like the power rule and subtraction rule to neatly combine terms into a more manageable form. In this exercise, after implementing the power rule and the subtraction rule, we condensed the expression \( \log_5 x - \frac{1}{5} \log_5 y \) into \( \log_5 \left( \frac{x}{y^{\frac{1}{5}}} \right) \).The idea is to express complex logarithmic statements into simpler, single-logarithm forms. This not only makes computations easier but often reveals insights or patterns that weren't obvious before. To simplify successfully:

• Look for opportunities to apply logarithm rules.
• Simplify step-by-step using known properties.
• Ensure that the base of all the logs involved match.

Simplifying is a valuable skill, especially when solving equations involving logarithms, and it forms a foundation for more advanced mathematical problem-solving.