Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions. $$\log (2 x+5)-\log x$$

Step-by-Step Solution

Answer

The condensed logarithmic expression is \(\log (2 + 5/x)\).

Step 1: Identify the Applicable Logarithmic Rule

The difference of two logarithms can be expressed as the logarithm of a ratio, \(\log a - \log b = \log (a/b)\). Hence, the given expression can be simplified using this property.

Step 2: Apply the Logarithmic Rule

By applying the rule to the given equation \(\log (2x+5) - \log x\), we get \(\log ((2x+5) / x)\). This is a single logarithmic expression with a coefficient of 1.

Step 3: Simplify the Expression Inside the Logarithm (Optional)

The expression inside the logarithm, \((2x+5) / x\), can be simplified to \(2 + 5/x\). Therefore, the final expression is \(\log (2 + 5/x)\). This can be the final step for this expression. However, without specific values for x, the expression can't be evaluated.

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is crucial for manipulating and simplifying logarithmic expressions. Logarithms, the inverse operations of exponentiation, share several properties that can make complex calculations more manageable.

• Product Property: The logarithm of a product is equal to the sum of the logarithms of the individual factors, formally \text{\( \log(a \cdot b) = \log(a) + \log(b) \)}.
• Quotient Property: The logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator, expressed as \text{\( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \)}.
• Power Property: The logarithm of a number raised to an exponent is the exponent times the logarithm of the number itself, or \text{\( \log(a^x) = x \cdot \log(a) \)}.

By applying these properties, you can transform logarithmic expressions into simpler forms that are often more convenient for solving equations and evaluating values.

Logarithmic Identities

Logarithmic identities are a set of equations that reflect the fundamental characteristics of logarithms. These identities are pivotal when simplifying complex expressions or solving logarithmic equations.

• The logarithm of 1 to any base is always 0, because any number raised to the power of 0 is 1: \text{\( \log_b(1) = 0 \)}.
• For any base b, the logarithm of the base itself is 1: \text{\( \log_b(b) = 1 \)}.
• The change of base formula allows one to compute logarithms in a different base: \text{\( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)} for any positive base c.

These identities are the backbone for working with logarithms across different bases and simplifying expressions into something more recognizable and calculable.

Simplifying Logarithms

Simplifying logarithmic expressions is a process that often involves applying the aforementioned properties and identities. The goal is to write a complex expression as a single logarithm with a coefficient of 1, to make it more understandable and easier to evaluate or further manipulate.

• Begin by identifying terms that can be combined using the product, quotient, or power properties.
• Break down the expression step by step, and apply the properties where they fit. Remember, practice makes perfect.
• Always look out for opportunities to utilize logarithmic identities to simplify the expression further.
• If given numeric values, you can evaluate the simplified expression to find a concrete solution.

When tackling an expression like \text{\( \log (2x+5) - \log x \)}, we apply the quotient property because it involves subtraction (difference of logarithms), leading us to \text{\( \log((2x+5)/x) \)}. Upon further simplification, we end up with \text{\( \log(2 + 5/x) \)}, a result that's easier to interpret and, if needed, to compute when the values are known.