Question

The value of \(\log \left(\frac{18}{14}\right)+\log \left(\frac{35}{48}\right)-\log \left(\frac{15}{16}\right)=\) (1) 0 (2) 1 (3) 2 (4) \(\log _{16} 15\)

Short answer

Question: Simplify the following logarithmic expression: $\log \left(\frac{18}{14}\right)+\log \left(\frac{35}{48}\right)-\log \left(\frac{15}{16}\right)$. Answer: (1) 0

Step-by-step solution

Write down the given expression

We are given the expression: $$\log \left(\frac{18}{14}\right)+\log \left(\frac{35}{48}\right)-\log \left(\frac{15}{16}\right)$$

Use the logarithm rules to simplify

We use the logarithm product rule, log(a) + log(b) = log(ab), then use the logarithm quotient rule, log(a) - log(b) = log(a/b) to simplify the given expression: $$\log \left(\frac{18}{14}\right)+\log \left(\frac{35}{48}\right)-\log \left(\frac{15}{16}\right) = \log \left(\frac{18}{14} \times \frac{35}{48} \div \frac{15}{16}\right)$$

Simplify the fractions

Multiply and divide the fractions as follows: $$\log \left(\frac{18}{14} \times \frac{35}{48} \div \frac{15}{16}\right) = \log \left(\frac{18 \times 35 \times 16}{14 \times 48 \times 15}\right)$$

Simplify the numbers

We can cancel out some terms in the numerator and denominator: $$\log \left(\frac{18 \times 35 \times 16}{14 \times 48 \times 15}\right) = \log \left(\frac{2\times 9 \times 5\times 7 \times 16}{2 \times 7 \times 3 \times 16 \times 15}\right) $$ Now, we cancel out the common factors: $$\log \left(\frac{2\times 9 \times 5\times 7 \times 16}{2 \times 7 \times 3 \times 16 \times 15}\right) = \log \left(\frac{9 \times 5}{3 \times 15}\right)$$

Simplify the remaining expression

Now, we simplify the remaining expression: $$\log \left(\frac{9 \times 5}{3 \times 15}\right) = \log \left(\frac{45}{45}\right) = \log(1)$$ We know from the properties of logarithm that \(\log(1)=0\). So, the value of the given expression is: $$\log \left(\frac{18}{14}\right)+\log \left(\frac{35}{48}\right)-\log \left(\frac{15}{16}\right) = 0$$ The correct answer is (1) 0.

Key concepts

Logarithm Properties

Logarithms are a fundamental part of mathematics used to solve equations involving exponentiation. They help in transforming multiplicative relationships into additive ones, making them easier to handle. Understanding logarithm properties is essential:

• Inverse Property: Logarithms are the inverse of exponentiation. For any positive numbers a and x, if a raised to a certain power equals x (i.e., \(a^y = x\)), then \( y \) is the logarithm of x with base a (i.e., \( \log_a(x) = y \)).
• Identity Property: The logarithm of 1, regardless of the base, is 0, because any number raised to the power of 0 is 1: \( \log_a(1) = 0 \).
• Positive Numbers: Logs are defined only for positive numbers, because a negative number cannot be raised to a power to get a real result.

In this exercise, using logarithms helped to transform a complex fraction multiplication into a simplified expression.

Logarithm Rules

To solve logarithmic equations, certain rules help simplify expressions:

• Product Rule: This rule states that \( \log(a) + \log(b) = \log(ab) \). It is handy when multiplying numbers inside a log.
• Quotient Rule: Similarly, \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \). This rule simplifies subtraction of logs into divisions within a single log.
• Power Rule: This rule illustrates that \( \log(a^b) = b\cdot \log(a) \), allowing for the exponent to be brought out front as a multiplier.

In this problem, the product and quotient rules were used to consolidate log expressions into a single, manageable logarithmic expression. This simplification simplified the calculation dramatically.

Fraction Simplification

Simplifying fractions is a crucial skill for mathematical problem-solving. In complex equations, like those involving logs, fraction simplification can be a major step in reducing complexity:

• Multiplying and Dividing: When dealing with fractions inside a log equation, multiply the numerators together and the denominators together.
• Cancel Common Factors: Break down numbers into their prime factors and cancel any common factors from the numerator and denominator.
• Reduce to Simplest Form: Always reduce to the simplest form to make solving the overall expression easier.

For the given problem, simplifying the fractions within the logs was paramount. It involved multiplying and dividing as well as canceling out the greatest common divisors to get a simple solution.

Mathematical Problem Solving

Mathematical problem solving involves breaking down complex expressions into small, manageable steps. Using a structured approach aids in simplifying and eventually solving equations:

• Identify the Problem: Start by understanding what the problem is asking. In this case, simplification of logarithmic expressions.
• Apply Relevant Techniques: Use mathematical techniques, such as logarithm rules and fraction simplification, to dismantle the problem into simpler parts.
• Perform Step-by-Step Simplification: Move through the problem incrementally, simplifying at each stage, aiming to make calculations easier.
• Check the Solution: Once simplified, verify each step to ensure accuracy, especially when dealing with logarithms where properties can sometimes lead to counterintuitive results.

In this logarithm exercise, identifying the effective use of log rules and simplifying fractions step by step was key. By addressing each part of the problem logically and methodically, the solution became evident.