Problem 95 - Algebra 2

Question

Use the properties of logarithms to write \(\log 12\) in four different ways. Name each property you use.

Step-by-Step Solution

Answer

\(\log 12\) can be rewritten in four different ways: \(\log 2 + \log 6\), \(\log 2 + \log 2 + \log 3\), \(2\log 2 + \log 3\), and \(\frac{\ln 12}{\ln 10}\)

Step 1: Express using Property 1: Product Rule

The product rule states: \(\log_b(mn) = \log_b m + \log_b n\)\nApplying this rule would allow us to express \(\log 12\) as: \(\log 2 + \log 6\) which is equivalent to \(\log(2*6)\)

Step 2: Express using Property 2: Product Rule

The product rule can be used multiple times if the number can be factorized more than once. In this case, \(12 = 2*2*3\). Therefore, we could apply the rule again to the same expression and get: \(\log 2 + \log 2 + \log 3\) which is equivalent to \(\log(2*2*3)\)

Step 3: Express using Property 3: Power Rule

The power rule states: \(\log_b(m^n) = n \log_b m\)\nLet's use this rule with one of the 2 values in our second expression: \(\log 2 + \log 2 + \log 3 = 2\log 2 + \log 3\)

Step 4: Express using Property 4: Change of Base Rule

The change of base rule states: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\) for any positive base \(c \neq 1\)\nUsing this rule, \(\log 12\) (which has an implied base 10) could be rewritten as \(\log 12 = \frac{\ln 12}{\ln 10}\). Here, we chose \(c = e\), hence the natural logarithm is used in the numerator and denominator

Key Concepts

Product Rule

The product rule is a vital property of logarithms and understanding it can simplify logarithmic expressions efficiently. The rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Imagine you have two numbers, say 2 and 6. Applying the product rule,

• Express the logarithm of their product: \( \log(2 \times 6) = \log 12 \).
• Then, write it as the sum of their logarithms: \( \log 2 + \log 6 \).

This demonstrates how a complex logarithm can be broken down into the sum of simpler terms. Your expression becomes easier to manage. Additionally, the product rule can be applied multiple times. For instance, if you break down 12 into more factors: \( 12 = 2 \times 2 \times 3 \). You could reapply the product rule to get:

• \( \log 12 = \log 2 + \log 2 + \log 3 \).

This method can be continued until the expression is in its most manageable form.

Power Rule

The power rule for logarithms relates to exponents and simplifies expressions where numbers are raised to powers. According to the power rule:

• \( \log_b(m^n) = n \log_b m \)

This means you can bring the exponent in front of the logarithm as a multiplier.Let's put this into practice with the expression \( \log 2 + \log 2 + \log 3 \), derived using the product rule earlier.Here, notice there are two logarithms of 2:

• \( \log 2 + \log 2 = 2 \log 2 \).

By applying the power rule, you consolidate repeated terms efficiently, making calculations simpler. This not only eases the complexity in expressions but also sharpens your equations, reducing potential errors when solving problems.

Change of Base Rule

The change of base rule is especially useful when dealing with logarithms of various bases. Often, we prefer to work with either the common logarithm (base 10) or the natural logarithm (base \(e\)). The change of base rule states:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

where \( c \) is any positive base except 1. Applying this rule allows you to convert logarithms to a base that makes calculations more convenient. For instance, you can rewrite \( \log 12 \) in terms of natural logarithms:

• \( \log 12 = \frac{\ln 12}{\ln 10} \).

This conversion is handy because many calculators have dedicated keys for natural and common logarithms. Thus, simplifying the way you perform calculations with different bases.

Logarithmic Expression

A logarithmic expression is simply a form that includes a logarithm. These expressions can appear complicated, but with the right properties, they can be simplified or transformed into easier formats for calculations. Consider the expression \( \log 12 \). By utilizing the product rule, power rule, and change of base rule, the expression can be:

• Broken down into a sum of smaller logarithmic terms (product rule).
• Condensed by consolidating repeated terms (power rule).
• Converted to another base for convenience (change of base rule).

Understanding these manipulations equips you to handle logarithms in equations and real-world problems. Simplifying logarithmic expressions is key to making calculations doable and results more accessible.