Problem 48 - Algebra and Trigonometry

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 without using a calculator. $$ \log (3 x+7)-\log x $$

Step-by-Step Solution

Answer

The expression condensed to a single logarithm with coefficient 1 is \(\log(3 + 7/x)\)

Step 1: Identify the pertinent property of logarithm

For this scenario, it is important to note the quotient rule for logarithms, given by: \(\log_a m - \log_a n = \log_a (m/n)\). This rule allows the subtraction of two logarithms to be formatted as the logarithm of a quotient.

Step 2: Apply the quotient rule

Apply the quotient rule to the given expression:\(\log (3x+7)-\log x = \log((3x+7)/x)\)

Step 3: Simplify the expression

Simplify the quotient inside the logarithm:\(\log ((3x+7)/x) = \log(3 + 7/x)\)

Key Concepts

Quotient Rule

The quotient rule for logarithms is a fundamental property used when simplifying expressions involving logarithms. It allows us to transform the difference of two logarithms into a single logarithm with a fraction inside. The rule states:\[\log_a m - \log_a n = \log_a\left(\frac{m}{n}\right)\]In this formula:

• \(m\) and \(n\) are values or expressions that you are taking the logarithm of.
• \(a\) is the base of the logarithms being used, which must be the same for both terms.

By applying the quotient rule, an expression that might seem complicated because of multiple logarithmic terms can be simplified into a compact form. This method is particularly useful for solving equations and simplifying calculations in mathematics. Remember that both \(m\) and \(n\) should be positive, as the logarithm of a non-positive number is undefined. Always ensure that your simplified expression maintains the domain of the original expression.

Single Logarithm

A single logarithm is the expression of multiple logarithms combined into one. It shows the power of using logarithmic properties to streamline complex expressions.
In the problem provided, we used the quotient rule to combine two logarithmic terms:

• \(\log(3x + 7)\)
• \(\log(x)\)

After applying the quotient rule, the expression transforms into:
\[\log\left(\frac{3x + 7}{x}\right)\]This method not only condenses the expression but also simplifies the visual representation and understanding of the problem. Reducing multiple logs to a single log makes it easier for further manipulation or solving related equations. Ultimately, expressing logarithms together in a simplified form can make them much more manageable.

Simplification of Expressions

Simplification is key in mathematics as it reduces complexity, making expressions easier to work with. After using the quotient rule, further simplification can often be achieved by reducing the terms inside the logarithm.
In our example, the expression \(\frac{3x+7}{x}\) can further be simplified by dividing each term by \(x\):
\[3 + \frac{7}{x}\]This step makes the expression simpler and more intuitive. Simplification helps in honing in on the essence of the mathematical expression by removing unnecessary complications.
This technique is especially useful in solving equations, as it often reveals solutions more clearly. Always take advantage of algebraic manipulations after applying logarithmic properties to ensure the expression is as simple as possible while still retaining all the necessary information.