Problem 65 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\log _{4} 7-\log _{4} 3=\log _{4} 4\)

Step-by-Step Solution

Answer

The statement is false.

Step 1: Apply logarithm rules

According to logarithm rules, \(\log _{b}(a) - \log_{b}(c) = \log_{b}(\frac{a}{c})\). Apply this rule to the given expression. So \(\log _{4} 7-\log _{4} 3\) will become \(\log_{4}( \frac{7}{3})\).

Step 2: Evaluate right-hand side expression

\( \log _{4} 4\) equals 1. This is because 4 to the power of 1 equals 4.

Step 3: Compare the two sides

From the first two steps, we get \(\log_{4}( \frac{7}{3}) = 1 \). This is incorrect because 4 to the power of 1 does not equate to \(\frac{7}{3}\). Therefore, the initial equation is incorrect.

Key Concepts

Logarithms

Logarithms are a way to solve problems involving exponential relationships in a more manageable fashion. In essence, a logarithm is the inverse operation of exponentiation. For instance, if you have a logarithmic equation like \(\log_b(a) = c\), this implies that \(b^c = a\). Understanding this relationship helps you perform various calculations involving growth patterns or exponential decay.Logarithms are not just numbers but are tools that allow us to transform multiplicative processes into additive ones, making complex calculations easier. There are different bases for logarithms, such as base 10 (common logarithm) and base \(e\) (natural logarithm), as well as any other positive number except 1. The choice of base depends on the context of the problem you're solving.

Logarithm Rules

To work efficiently with logarithmic expressions, certain rules have been established based on the properties of exponents. Knowing these rules can simplify many complex expressions. Here are some of the most fundamental logarithm rules:

• Product Rule: \(\log_b(MN) = \log_b M + \log_b N\). This means when you multiply two numbers, you add their logarithms.
• Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b M - \log_b N\). Dividing translates to subtracting logarithms. This is the rule used in the exercise to simplify the expression \(\log_4 7 - \log_4 3\).
• Power Rule: \(\log_b(M^n) = n\log_b M\). This states that a logarithm with a power can be simplified by bringing the exponent in front of the logarithm.
• Change of Base Formula: \(\log_b M = \frac{\log_k M}{\log_k b}\). It is useful when you need to switch between different logarithmic bases.

Using these rules allows for manipulation and simplification, making it easier to solve equations and analyze logarithmic expressions.

Logarithmic Equations

Logarithmic equations involve expressions with logarithms and require specific strategies to solve. The goal is often to isolate the logarithm on one side and then exponentiate to find the variable.To tackle logarithmic equations effectively, follow these general steps:

• Apply logarithmic rules to combine or separate terms. This can involve both the product and quotient rules.
• Try to simplify the equation so that it contains a single logarithm. This will make it easier to remove the logarithm using exponentiation.
• Exponentiate both sides of the equation to eliminate the logarithm, allowing you to solve for the unknown variable.

Apply these steps carefully to ensure accuracy. For the specific exercise given, we observed that by using the quotient rule, we transformed \(\log_4 7 - \log_4 3\) into \(\log_4(\frac{7}{3})\). Comparing this with the right-side \(\log_4 4\) equal to 1, we saw they were not equivalent. Thus, the equation proved false by evaluating each side independently.