Problem 71 - Algebra 2

Question

Write each logarithm as the quotient of two common logarithms. Do not simplify the quotient. $$ \log _{9} 3.3 $$

Step-by-Step Solution

Answer

The logarithm \(\log _{9} 3.3\) as a quotient of two common logarithms is \(\frac{\log 3.3}{\log 9}\).

Step 1: Apply the change of base formula

The change of base formula states: \(\log_b a = \frac{\log a}{\log b}\), where \(\log\) denotes the common (base 10) logarithm.

Step 2: Apply to the given expression

\(\log_9 3.3 = \frac{\log 3.3}{\log 9}\)

Key Concepts

Change of Base Formula

Logarithms can seem tricky at first, but with the right formulas, they become much easier to handle. One significant tool in the logarithm toolkit is the Change of Base Formula. This formula is essential when you want to evaluate logarithms that aren't based on 10 or 'e', the natural logarithm base. Often, calculators don't have a direct option for bases other than these standard ones.

The Change of Base Formula states:

• \[\log_{b} a = \frac{\log_{c} a}{\log_{c} b}\]
• This means you can change any logarithm to a different base using the base you choose. Any base can be used for 'c', but it's common to use base 10 (common logarithms) or base 'e' (natural logarithms).
• This is particularly useful when solving exercises where the base of the log needs to be simplified or evaluated using standard calculators.

For example, in the original exercise we have \(\log _{9} 3.3\). By applying the formula with common logarithms (base 10), it becomes \(\frac{\log 3.3}{\log 9}\). Remember that the numerator is the log of the argument (3.3), and the denominator is the log of the base (9). This transformation simplifies the calculation process and is a helpful way to manage more complex logarithmic expressions.

Common Logarithms

When we talk about common logarithms, we're referring specifically to logarithms with a base of 10. They are called 'common' because they are frequently used in various mathematical and real-life applications.

The notation for a common logarithm simplifies to \(\log a\), where it's inherently understood that the base is 10. You might wonder why base 10? Well, it's because it's relatively simple for manual calculations and aligns with our usual number system.

• Common logarithms can be easily calculated with most scientific calculators, making them very handy in both academic and practical settings.
• They're used in fields such as engineering, finance, and any discipline involving exponential growth or decay.

In practice, common logarithms serve as the 'silent helpers' behind many conversion calculations like the one in the exercise. By converting logarithms like \(\log _{9} 3.3\) into common logarithms, you can work with familiar bases and tools to solve more complex logarithmic equations.

Quotient of Logarithms

Understanding the quotient of logarithms is a key part of effectively using the Change of Base Formula. When we express a logarithm as a quotient, we are essentially using the properties of logarithms to separate different components of the original log expression into a ratio.

For example, when you see \(\frac{\log 3.3}{\log 9}\), you're seeing how the original log statement \(\log _{9} 3.3\) is rearranged using the chain of logs divided by each other:

• The numerator \(\log 3.3\) represents the log of the number we're evaluating.
• The denominator \(\log 9\) represents the log of the base that we are initially working with.

The quotient structure provides a powerful simplification, allowing for calculations that incorporate different bases without needing to change overall logarithmic relationships. This setup is particularly beneficial in making the computation process easier with standard tools.

By calculating each component separately, you're breaking down potentially complex exponential expressions into manageable bits. It’s a bit like translating a language you don't speak into one you understand fluently, piece by piece. This method not only clarifies the calculations needed but also highlights the relationship between the numbers involved.