Problem 80 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. $$ 3 \log 2-\log 4 $$

Step-by-Step Solution

Answer

The simplified expression is \(\log(2)\).

Step 1: Apply Power Rule

The first step is to apply the power rule, \(a \cdot \log_b(x) = \log_b(x^a)\), to the term \(3 \log 2\), meaning the 3, which is a coefficient of the logarithm, will become the exponent of the number 2. This yields \(\log(2^3) - \log 4\).

Step 2: Simplify

Then, simplify the expression \(\log(2^3)\) to get \(\log(8) - \log(4)\).

Step 3: Apply Division Rule

Finally, apply the division rule for logarithms, \(\log_c(b) - \log_c(a) = \log_c(\frac{b}{a})\), to get a single logarithm. This results in the final answer being \(\log(\frac{8}{4})\).

Key Concepts

Power Rule

The power rule is a crucial concept in logarithmic operations which states that a coefficient multiplying a logarithm can be repositioned as an exponent of the argument. In formula terms, this rule is expressed as \(a \cdot \log_b(x) = \log_b(x^a)\). It essentially helps in simplifying expressions by transforming the coefficient into the power of the base of the logarithm.

For instance, in the expression \(3 \log 2\), the number 3, which is a coefficient, is moved as an exponent of 2. This power rule transforms \(3 \log 2\) into \(\log(2^3)\). This step is fundamental as it lays the groundwork for further simplification and application of other logarithmic rules.

Division Rule

Logarithms allow us to simplify expressions using the division rule. The division rule states that when subtracting two logarithms with the same base, you can create a single logarithm with the fraction of the two arguments. It is stated as \(\log_c(b) - \log_c(a) = \log_c\left(\frac{b}{a}\right)\).

After applying the division rule to \(\log(8) - \log(4)\), you get \(\log\left(\frac{8}{4}\right)\). The subtraction of the logs simplifies into a single logarithm by taking the division of their arguments (8 divided by 4). This powerful simplification reduces complex expressions and aids in solving equations more easily.

Simplifying Logarithms

Simplifying logarithms involves using the rules, like the power and division rules, to condense expressions into a more manageable form. This process makes solving logarithmic expressions more straightforward.

In the given exercise, the steps illustrate how to simplify \(3 \log 2 - \log 4\) into \(\log\left(\frac{8}{4}\right)\), which further simplifies to \(\log(2)\). Simplifying is all about transforming the terms to become more unified, often by reducing the number of operations needed to reach a single logarithm.

• Applying the power rule changes the expression by reconfiguring coefficients into exponents.
• The division rule then transforms a subtraction of logs into a single logarithm of a simplified fraction.

Through this systematic approach, complexities are reduced, making it easier to handle logarithmic equations.