Problem 53 - Algebra 2 and Trigonometry

Question

In \(48-55,\) if \(\log a=c,\) express each of the following in terms of \(c\) $$ \log \frac{a^{2}}{10} $$

Step-by-Step Solution

Answer

The expression is \(2c - 1\).

Step 1: Apply Logarithm Properties

Using the property of logarithms that states \( \log \frac{m}{n} = \log m - \log n \), we rewrite the expression \( \log \frac{a^2}{10} \) as \( \log a^2 - \log 10 \).

Step 2: Expand Using Power Rule

Use the power rule for logarithms which states \( \log m^n = n \cdot \log m \) to expand \( \log a^2 \). This gives us \( 2 \log a \).

Step 3: Substitute Known Values

Substitute \( \log a = c \) into the expression. So, \( 2 \log a \) becomes \( 2c \). Substitute this into the previous step: \( 2c - \log 10 \).

Step 4: Substitute \(\log 10\) Value

We know that \( \log 10 = 1 \) because \( 10 \) is the base of the common logarithm. Substitute this into the expression to get \( 2c - 1 \).

Key Concepts

Logarithm Properties

Logarithm properties serve as rules that help us manipulate and simplify logarithmic expressions. These properties make certain difficult calculations accessible by transforming complex logs into simpler forms. One crucial property is the quotient rule of logarithms, which allows us to handle division inside a log. If you have a logarithmic expression like \( \log \frac{m}{n} \), you can rewrite it using the rule:

• \( \log \frac{m}{n} = \log m - \log n \)

This makes it easier to perform operations by breaking the expression into a subtraction of two separate logarithms. Applying these types of properties systematically can effectively solve logarithmic equations and simplify expressions.

Power Rule for Logarithms

The power rule for logarithms is a powerful tool when you need to work with exponents inside a logarithmic expression. This rule states that if you have an expression like \( \log m^n \), you can simplify it to:

• \( \log m^n = n \cdot \log m \)

Using this rule, you take the exponent \( n \) out of the logarithm, turning it into a coefficient in front of \( \log m \). This is particularly helpful when dealing with expressions where the base raised to a power might otherwise be challenging to handle. The power rule streamlines the calculation, helping you focus on the main components of your logarithmic problem.

Logarithmic Substitution

Logarithmic substitution is a method used to simplify equations, especially when a particular logarithm is related to a known value. For example, if you know \( \log a = c \), substitution allows you to replace \( \log a \) with \( c \) in any given expression.

• When you see \( 2 \log a \), and you know \( \log a = c \), simply substitute to obtain \( 2c \).

By replacing the parts of the logarithmic expression with what you already know, the equation becomes less complex. This approach is extremely valuable in problems where substituting gives a direct path to the solution.

Common Logarithm

The common logarithm, denoted as \( \log \), is a logarithm with base 10. It’s widely used due to its simplicity and because it's the base of the metric system. An essential aspect of the common logarithm is that \( \log 10 = 1 \). This is because raising 10 to the power of 1 results in 10.

• In any logarithmic expression, if you see \( \log 10 \), you can substitute it with 1, making calculations straightforward.

Understanding this property is crucial. It simplifies expressions and is based on the fundamental definition of a logarithm: finding the power to which a base must be raised to produce a given number. Recognizing and utilizing the common logarithm makes working with logs quicker and easier.