Problem 52 - Algebra 2 and Trigonometry

Question

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \frac{100}{a} $$

Step-by-Step Solution

Answer

\( \log \frac{100}{a} = 2 - c \)

Step 1: Apply the Quotient Rule of Logarithms

Recall that the logarithm of a quotient can be expressed as the difference of two logarithms. Apply the quotient rule here: \( \log \frac{100}{a} = \log 100 - \log a \).

Step 2: Simplify the Logarithm of 100

The base-10 logarithm of 100 is 2 because \( 10^2 = 100 \). Therefore, \( \log 100 = 2 \). Substituting this into the equation gives us \( \log \frac{100}{a} = 2 - \log a \).

Step 3: Substitute the Given Value for \( \log a \)

We know from the problem that \( \log a = c \). Substitute \( c \) for \( \log a \) in the equation: \( \log \frac{100}{a} = 2 - c \).

Key Concepts

Quotient Rule of Logarithms

In logarithms, the quotient rule is a handy tool useful when dealing with division inside a logarithmic function. This rule helps simplify the logarithm of a fraction into an easier form.
The quotient rule states:

• If you have a logarithm of a fraction, for instance, \( \log \left( \frac{b}{a} \right) \), you can express it as the difference of the logarithms: \( \log b - \log a \).

In the original exercise, the problem involved \( \log \left( \frac{100}{a} \right) \). By using the quotient rule, it was split into \( \log 100 - \log a \). This step turns a complex expression into a simpler one. Remembering the rule as "a logarithm of a quotient equals the difference of logs" will help you deal efficiently with division inside logs.

Logarithmic Identities

Logarithmic identities are basic formulas that make it easier to work with logarithms. These identities are foundational tools for simplifying expressions and solving equations.
Here are a few key logarithmic identities:

• Product Rule: \( \log(ab) = \log a + \log b \).
• Quotient Rule: \( \log \left( \frac{b}{a} \right) = \log b - \log a \), which was used in the original problem.
• Power Rule: \( \log(a^b) = b \log a \).
• Logarithm of 1: \( \log 1 = 0 \) because any number to the power of 0 is 1.
• Base Change Formula: \( \log_b a = \frac{\log_c a}{\log_c b} \).

These identities serve as shortcuts and eliminate complex steps in calculations. In the exercise, one key identity used was finding \( \log 100 = 2 \), by recognizing that \( 10^2 = 100 \). Knowing these rules allows you to tackle logarithmic expressions with confidence and ease.

Expressing Logarithms in Terms of Variables

The exercise's goal was to express a given logarithm, \( \log \left( \frac{100}{a} \right) \), using a specific variable. When dealing with variables, it’s important to substitute them precisely once expressions are simplified.
In the original problem, it was given that \( \log a = c \). This information is essential. Through substitution, known values convert complex expressions to simpler forms. After applying the quotient rule and simplifying \( \log 100 \), the problem boiled down to \( \log \left( \frac{100}{a} \right) = 2 - \log a \). Then, simply substitute \( \log a \) with \( c \).
Ultimately, the solution became \( 2 - c \). Practicing expressing complex logarithmic expressions using variables reinforces understanding of substitution, and it aids in solving more intricate log problems quickly.