Problem 18 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. \(\log _{7} x+\log _{7} y-\log _{7} z\)

Step-by-Step Solution

Answer

The simplified form of \( \log _{7} x + \log _{7} y - \log _{7} z \) is \( \log _{7} \left(\frac{x \cdot y}{z}\right) \).

Step 1: Apply Addition Property of Logarithm

Use the addition property of logarithm to write \( \log _{7} x + \log _{7} y \) as \( \log _{7} (x \cdot y) \). This results in \( \log _{7} (x \cdot y) - \log _{7} z \).

Step 2: Apply Subtraction Property of Logarithm

Use the subtraction property of logarithm to write \( \log _{7} (x \cdot y) - \log _{7} z \) as \( \log _{7} \left( \frac{x \cdot y}{z} \right) \).

Key Concepts

Addition Property of Logarithms

When you see an expression like \( \log_b a + \log_b b \), you can simplify it using the addition property of logarithms. This property allows us to combine two logarithms with the same base into a single log expression. It states that:

• \( \log_b a + \log_b b = \log_b (a \cdot b) \)

This means that when you add two logarithms that share the same base, you can multiply their respective arguments. So, for \( \log_7 x + \log_7 y \), you would combine them to form \( \log_7 (x \cdot y) \).
This is incredibly useful because it lets us simplify expressions that have multiple log terms into fewer terms.
This simplification can make it easier to further manipulate or solve equations involving logarithms.

Subtraction Property of Logarithms

Subtraction in logarithmic expressions also allows for simplification using a specific property. The subtraction property of logarithms lets us express the difference of two logs with the same base as a single logarithm:

• \( \log_b a - \log_b b = \log_b \left( \frac{a}{b} \right) \)

In this context, if you start with \( \log_7 (x \cdot y) - \log_7 z \), you can condense it into a single term by dividing \( x \cdot y \) by \( z \).

This results in \( \log_7 \left( \frac{x \cdot y}{z} \right) \). This approach reduces the number of terms and often simplifies solving or further manipulation of logarithmic equations.

Single Logarithm

The idea of expressing multiple log terms as a single logarithm is a powerful tool in mathematics. Once you understand and apply both the addition and subtraction properties of logarithms, you're able to reduce complex expressions into simpler, condensed forms. A single logarithm expression, like \( \log_7 \left( \frac{x \cdot y}{z} \right) \), includes all the components of the original log expression, but in a more straightforward form.

Using these properties:

• You simplify calculations.
• Make it easier to address equations.
• Can reveal deeper insights into the relationships between the variables and constants involved.

Ultimately, mastering single logarithm expressions allows you to handle a wide range of logarithmic problems with confidence and clarity.