Problem 48 - College Algebra

Question

In Exercises \(41-70,\) 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 simplified single logarithm for the given expression is \(\log(3+\frac{7}{x})\)

Step 1: Identify the Logarithmic Property

We start by identifying the property of logarithm we will use to solve the problem. In this case the property is \(\log _{b}(M)-\log _{b}(N)=\log _{b}(\frac{M}{N})\) which allows us to condense the subtraction of two logarithms into a division within a single logarithm.

Step 2: Apply the Logarithmic Property

Now, apply this property to the expression we have. Substituting the given expression \(\log(3x+7) - \log(x)\) using the property, it turns into \(\log \left( \frac{3x+7}{x} \right)\).

Step 3: Simplify the Expression

The final step is to simplify the expression within the logarithm. As the divisor is \(x\), divide each term in the numerator by \(x\). This gets us to the final simplified single logarithmic expression: \(\log \left( \frac{3 + \frac{7}{x}}{1} \right) \) or simply \(\log(3+\frac{7}{x})\).

Key Concepts

Understanding Logarithmic Expressions

Logarithmic expressions are mathematical statements involving logarithms, which are inverse operations to exponentials. They help us manage and simplify expressions where numbers are raised to certain powers. A logarithmic expression takes the form \(\log_b(M)\), where \(b\) is the base, and \(M\) is the argument or the number for which we are finding the logarithm. In simpler terms, it answers the question: "To what power must the base be raised, to yield the argument?" For example, \(\log_2(8) = 3\) because \(2^3 = 8\).

• Base of Logarithms: This can be any positive number, but common bases include 10 (common logarithm), \(e\) (natural logarithm), and 2 (binary logarithm).
• Argument: The number inside the logarithm that we are interested in.

Understanding logarithmic expressions allows us to work with these numbers more intuitively, and perform operations like addition, subtraction, multiplication, and division through logarithmic properties.

Exploring Logarithmic Property

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. These properties allow us to condense or expand logarithms, making complex calculations more manageable. One such property is the quotient rule: \(\log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right)\). This property condenses the subtraction of two logarithms into a single logarithm of a quotient.

• Quotient Rule: As you've seen, this is useful for simplifying expressions by turning a subtraction into a division.
• Product Rule: \(\log_b(M) + \log_b(N) = \log_b(M \cdot N)\), used for condensing sums into products.
• Power Rule: \(n \cdot \log_b(M) = \log_b(M^n)\), helps with consolidating multiplicative coefficients.

These properties are fundamental tools in mathematics as they enable us to simplify and evaluate complex expressions efficiently, especially in algebra and calculus.

Condensing Logarithms

Condensing logarithms involves using logarithmic properties to rewrite multiple logarithmic terms into a single term. This simplification is effective for solving equations, making integrals easier to handle, or simplifying complex expressions for further analysis. In the original exercise, we condensed \(\log(3x+7) - \log(x)\) into \(\log\left(\frac{3x+7}{x}\right)\).

• Prerequisites for Condensation: Ensure both logarithms have the same base, which is typically implied.
• Simplification Strategy: Apply the quotient rule of logarithms to merge terms into a single logarithmic expression.

Once condensed, you can further simplify the argument of the logarithm. For example, dividing terms within the logarithm results in \(\log(3+\frac{7}{x})\). Condensing logarithms is not just about simplifying—it is about transforming the expression into a potentially more useful form for further mathematical operations or understanding.