Question

The value of \(\log (a b)^{2}-\log (a c)+\log \left(b c^{4}\right)-3 \log (b c)\) is : (a) 0 (b) \(\log b\) (c) \(\log c\) (d) \(\log a\)

Short answer

Answer: None of the given answer choices match the simplified expression. The correct answer should be \(\log\left(\frac{ab}{c^2}\right)\). The options provided might be incorrect or the exercise might have some errors.

Step-by-step solution

Recall the properties of logarithms

We need to remember two properties of logarithms: 1. \(\log (x * y) = \log x + \log y\) (Product Rule) 2. \(\log (x^y) = y * \log x\) (Power Rule) Now let's apply these properties to simplify the given expression.

Apply the power rule

Rewrite the expression by applying the power rule to the first term: \(\log (a b)^{2}-\log (a c)+\log (b c^4)-3 \log(b c) = 2*\log(ab) - \log(ac) + \log(bc^4) - 3*\log(bc)\)

Apply the product rule

Rewrite the expression by applying the product rule to each term: \(2*\log(ab) - \log(ac) + \log(bc^4) - 3*\log(bc) = 2(\log a + \log b) - (\log a + \log c) + (\log b + 4\log c) - 3(\log b + \log c)\)

Expand and simplify the expression

Expand the expression and group the like terms together: \((2\log a + 2\log b) - (\log a + \log c) + (\log b + 4\log c) - (3\log b + 3\log c)\) Now, let's simplify: \(\log a + \log b - 2\log c\)

Apply the product rule in reverse

We can express the simplified expression using the product rule in reverse: \(\log a + \log b - 2\log c = \log(ab) - \log(c^2)\)

Apply the quotient rule

We can now apply the quotient rule in reverse: \(\log(ab) - \log(c^2) = \log\left(\frac{ab}{c^2}\right)\) Now let's compare this expression with the answer choices: (a) 0 (b) \(\log b\) (c) \(\log c\) (d) \(\log a\) None of the answer choices match the simplified expression, which means there might be a mistake in the given options or the exercise might have some errors.

Key concepts

Properties of Logarithms

Understanding the properties of logarithms is essential when dealing with logarithmic expressions. These properties provide rules that help simplify and solve complex logarithm problems.

• Product Rule: The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as: \[\log(xy) = \log x + \log y\]
• Power Rule: The power rule states that the logarithm of a power of a number is equal to the exponent times the logarithm of the base. This is expressed as: \[\log(x^y) = y \log x\]
• Quotient Rule: The quotient rule is used when you have the logarithm of a division and expresses it as the difference of the logarithms: \[\log\left(\frac{x}{y}\right) = \log x - \log y\]

Recognizing these properties helps manipulate and simplify logarithmic expressions efficiently.

Product Rule

The product rule for logarithms is particularly useful when you encounter expressions that involve the multiplication of variables or numbers under a logarithmic function. This rule helps by breaking down large, complex logarithms into smaller, more manageable parts.

Using the Product Rule

When applying the product rule, you take the logarithm of two numbers multiplied together and convert it into the sum of two separate logarithms. As an example: \[\log(2 \cdot 3) = \log 2 + \log 3\]
This property is inversely related as well, meaning the sum of two logarithms can be rewritten as a single logarithm of a product. For instance: \[\log 2 + \log 3 = \log(2 \cdot 3)\] Understanding this concept allows you to simplify expressions where multiplication is present inside a logarithmic expression, making them easier to calculate or further manipulate.

Power Rule

The power rule is a key tool when you deal with expressions that involve exponential expressions within a logarithm. It directly relates the concept of power to logarithms, simplifying calculations significantly.

Applying the Power Rule

This rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of that number: \[\log(x^n) = n \log x\]
For instance, if you have a logarithm such as \(\log(5^3)\), by applying the power rule, it simplifies to: \(3 \cdot \log 5\).
This drastically reduces the complexity of an expression, allowing for easier manipulation and simplification. The power rule is particularly useful when combined with the product and quotient rules to untangle complicated logarithmic equations.

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions involves applying the properties of logarithms to condense a complex expression into a simpler form. This process is crucial for solving logarithmic equations effectively.

Steps to Simplify Logarithmic Expressions

• Apply the Power Rule: First, handle any powers inside logarithms by moving them in front of the logarithm using the power rule. For example: \(\log(x^4) = 4 \log x\).
• Use the Product Rule: Apply the product rule to break down logarithms of products into sums: \(\log(xy) = \log x + \log y\).
• Apply the Quotient Rule: Convert expressions involving division within a logarithm into subtraction: \(\log\left(\frac{x}{y}\right) = \log x - \log y\).
• Combine Like Terms: Simplify the expression by combining logarithmic terms wherever applicable, reducing clutter and making it easier to solve.

Using these steps systematically will render a complicated logarithm problem into a straightforward one, paving the way for easier problem-solving.