Problem 7 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{7}\left(\frac{7}{x}\right) $$

Step-by-Step Solution

Answer

The expanded and simplified form of the logarithmic expression is \(1 - \log_7(x)\)

Step 1: Apply the Quotient Rule

To start with, we can apply the quotient rule of logarithms to the given expression \(\log _{7} \left(\frac{7}{x}\right)\). The quotient rule states that \(\log_b (M/N) = \log_b(M) - \log_b(N)\), where b, M, and N are positive real numbers, M ≠ 1 and N ≠ 0. Here, b is 7, M is 7, and N is x. Substituting these into the rule gives us \(\log_7(7) - \log_7(x)\)

Step 2: Simplify \(\log_7(7)\)

Since the base of the logarithm and the argument are the same number (7 in this case), the value of the logarithm is 1 according to the rule \(\log_b (b) = 1\). So, \(\log_7(7) = 1\). Substitute this into the expression from Step 1 to get \(1 - \log_7(x)\)

Step 3: Final Answer

So, the logarithmic expression \(\log _{7} \left(\frac{7}{x}\right)\) expands and simplifies to \(1 - \log_7(x)\)

Key Concepts

Logarithmic Expressions

Logarithmic expressions are a way to represent exponents in a compact form. When you see a logarithm, it typically looks like \(\log_b(a)\), where "\(b\)" is the base, and "\(a\)" is the argument, or the number you are taking the logarithm of. The equation \(\log_b(a) = c\) implies that \(b^c = a\). This is the logarithm solving for "\(c\)", which is the exponent needed to raise the base "\(b\)" to produce "\(a\)".

Understanding how to work with logarithmic expressions requires familiarity with various rules and properties, which allow for simplifying complex expressions. Simplification can help solve equations or make calculations easier. Recognizing parts of expressions, like the quotient in \(\log_7\left(\frac{7}{x}\right)\), is key to applying rules efficiently. This approach breaks complicated expressions down into more manageable pieces.

Properties of Logarithms

Logarithms have several important properties that help in manipulating and simplifying expressions. These properties arise from the nature of exponents and include:

• The Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\). This rule applies when multiplying inside the logarithm, allowing the separation into a sum of logs.
• Power Rule: \(\log_b(M^p) = p \cdot \log_b(M)\). Using this rule, any exponent inside a logarithm can be moved in front as a factor.
• Change of Base Formula: Often useful for computing logs with arbitrary bases; allows you to convert a logarithm to a different base.

These properties can make logarithms much simpler and easier to handle, as they allow breaking single expressions into parts. For example, you can apply these rules step by step to elements within your logarithmic expression until it’s as simple as possible.

Quotient Rule of Logarithms

The quotient rule for logarithms is a handy tool for dealing with fractions inside the argument of a logarithm. This rule states that \(\log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N)\). What this rule means is, whenever you have a division within a logarithm, you can separate it into the difference between two logarithms.

In our example \(\log_7 \left( \frac{7}{x} \right)\), we see this rule at work: the fraction under the logarithm is split into \(\log_7(7) - \log_7(x)\). This can make complex expressions much simpler to evaluate. Knowing this rule not only helps in logarithmic simplifications, but it also offers insight into how logarithms provide a toolkit for solving real-world problems involving ratios or divisions.