Problem 28 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ 2 \log _{7} y+6 \log _{7} z $$

Step-by-Step Solution

Answer

\( \log_7 (y^2 \cdot z^6) \)

Step 1: Apply the Power Rule

The power rule of logarithms states that \( a \log_b x = \log_b (x^a) \). Apply this rule separately to each term. - For the first term: \( 2 \log_7 y = \log_7 (y^2) \).- For the second term: \( 6 \log_7 z = \log_7 (z^6) \).

Step 2: Apply the Product Rule

The product rule of logarithms states that \( \log_b x + \log_b y = \log_b (xy) \). Now, applying this rule to the results from Step 1:- Combine \( \log_7 (y^2) + \log_7 (z^6) \) into a single logarithm: \( \log_7 (y^2 \cdot z^6) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is quite straightforward but incredibly powerful. It allows you to bring down the exponent of a number in your logarithmic expression. This rule states:

• \( a \log_b x = \log_b (x^a) \).

By using this rule, you can simplify expressions like \( 2 \log_7 y \) into \( \log_7 (y^2) \). Notice here how we transformed the coefficient 2 into an exponent of the variable \( y \).
This rule is particularly handy when you deal with products or quotients of logarithms, as it sets up your equation to apply other logarithmic rules.
For every coefficient in front of a logarithm, think of it in terms of an exponent inside the logarithm. Remember, this only works because logarithms are exponents, so you can manipulate them in similar ways.

Product Rule of Logarithms

When working with multiple logarithmic expressions added together, the product rule of logarithms comes to the rescue. According to this rule:

• \( \log_b x + \log_b y = \log_b (xy) \).

What this means is you can turn the sum of two logarithms into a single logarithm by multiplying the values inside. For instance, given \( \log_7 (y^2) + \log_7 (z^6) \), you can consolidate this into \( \log_7 (y^2 \cdot z^6) \).
This is especially useful because having a single log expression is often much simpler to manipulate.
Using this rule helps combine terms efficiently and helps in solving more complex logarithmic equations without getting overwhelmed with too many individual pieces.

Properties of Logarithms

Understanding the key properties of logarithms is crucial when learning how to simplify logarithmic expressions. Here are some of the important properties:

• **Product Property:** \( \log_b (xy) = \log_b x + \log_b y \).
• **Quotient Property:** \( \log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y \).
• **Power Property:** \( \log_b (x^a) = a \log_b x \).
• **Change of Base Formula:** \( \log_b (x) = \frac{\log_k (x)}{\log_k (b)} \). You can use this to change the base of a logarithm.

These properties allow you to manipulate and solve equations involving logarithms by converting between different forms.
Being familiar with them gives you more flexibility in simplifying and solving complex problems, making the expression easier to work with.
Each property opens up new ways to approach and solve log-related questions by transforming the expression into a more manageable form.