Problem 50 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \log 100a = 2 + c \)

Step 1: Use Logarithm Product Rule

The logarithmic expression \( \log 100a \) can be separated into two components using the product rule of logarithms, which states \( \log(xy) = \log x + \log y \). We can rewrite the expression as \( \log 100 + \log a \).

Step 2: Determine \( \log 100 \)

The value of \( \log 100 \) can be determined by recognizing that 100 is \( 10^2 \). Therefore, \( \log 100 = \log (10^2) \). Using the power rule for logarithms, \( \log (10^2) = 2 \log 10 \). Since \( \log 10 = 1 \), we have \( 2 \times 1 = 2 \).

Step 3: Substitute \( \log a = c \)

We know from the problem statement that \( \log a = c \). Substitute \( c \) for \( \log a \) in the expression from Step 1. We now have \( \log 100a = \log 100 + c \).

Step 4: Combine the Expressions

Substituting the calculated value \( \log 100 = 2 \) into the expression, it becomes \( 2 + c \). Thus, \( \log 100a \) expressed in terms of \( c \) is \( 2 + c \).

Key Concepts

Logarithm Product Rule

The logarithm product rule is a key concept when dealing with logarithmic expressions. This rule allows us to simplify the expression of the logarithm of a product into a sum of logarithms. Specifically, the product rule is expressed as:

• \[ \log(xy) = \log x + \log y \]

If you have an expression like \( \log 100a \), you can separate it using the product rule. This will break it down into \( \log 100 + \log a \). This simplification is particularly useful when both \( x \) and \( y \) are numbers or known expressions, allowing for easier manipulation and calculation. Always remember this rule when you encounter products within a logarithm. It can significantly simplify your problem-solving process by converting products into more manageable additions.

Power Rule for Logarithms

The power rule for logarithms is another important tool. It helps simplify log expressions where the argument is an exponentiation of a base. According to the power rule:

• \[ \log(x^n) = n \log x \]

This means that if you have a power within the logarithm, you can "bring down" the exponent as a multiplier of the log. For example, consider \( \log(10^2) \). Applying the power rule, you get \( 2 \log 10 \). This transformation simplifies the evaluation of power-heavy expressions, as you often know the values of common logarithms like \( \log 10 \), which equals 1. Utilizing this rule, established powers can be isolated and managed separately, assisting in clearer and more straightforward calculations, especially when combined with other log rules like the product rule.

Substitution in Logarithms

Substitution in logarithms is a powerful method used to replace complex components of a logarithmic expression with simpler ones or with provided values. In scenarios where you are given a known value, like \( \log a = c \), substitution helps solve for other expressions in terms of that known value. For instance, in the expression \( \log 100a \), once you express the log as \( \log 100 + \log a \), you can substitute \( \log a \) with \( c \), simplifying the problem to \( \log 100 + c \). This reduction is essential because the expression is now in terms of \( c \), which simplifies calculations or further problem-solving steps.Consistent application of substitutions makes resolving logarithmic equations easier. It allows you to maintain coherent expressions in terms of given values, making complex logarithmic computations much more manageable.