Problem 44 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{10^{x}}{x\left(x^{2}+1\right)\left(x^{4}+2\right)}\right) $$

Step-by-Step Solution

Answer

The expanded form is \(x\log(10) - \log(x) - \log(x^2+1) - \log(x^4+2)\).

Step 1: Apply the Quotient Rule

To begin expanding the logarithmic expression, we apply the Quotient Rule of logarithms. The Quotient Rule states that \( \log \left(\frac{a}{b}\right) = \log(a) - \log(b) \). Thus, we have:\[ \log \left( \frac{10^{x}}{x\left(x^{2}+1\right)\left(x^{4}+2\right)} \right) = \log(10^x) - \log\left(x\left(x^2+1\right)\left(x^4+2\right)\right).\]

Step 2: Expand the Numerator

Focus on the numerator, \(10^x\). The logarithm of an exponential function is simplified using the Power Rule of logarithms: \(\log(a^b) = b\log(a)\). Hence,\[\log(10^x) = x \cdot \log(10).\]

Step 3: Apply the Product Rule

Now, apply the Product Rule to the expression in the denominator, \(x(x^2+1)(x^4+2)\). The Product Rule is \(\log(abc) = \log(a) + \log(b) + \log(c)\). Therefore, we have:\[\log\left(x(x^2+1)(x^4+2)\right) = \log(x) + \log(x^2+1) + \log(x^4+2).\]

Step 4: Combine and Simplify the Expression

Insert the expressions from Steps 2 and 3 back into the equation from Step 1:\[\log(10^x) - \left(\log(x) + \log(x^2+1) + \log(x^4+2)\right).\]Distribute the negative sign:\[x\log(10) - \log(x) - \log(x^2+1) - \log(x^4+2).\]

Key Concepts

Quotient Rule

In the study of logarithms, the Quotient Rule is a fundamental property. This rule helps us manage the division of two values inside a logarithm. To simplify this, the rule states that the logarithm of a quotient is equal to the difference of the logarithms: \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \). For example, if you have a complex logarithmic expression where there is a division in the argument, the Quotient Rule lets you split the expression into two separate logs. Consider the expression \( \log \left( \frac{10^x}{x(x^2+1)(x^4+2)} \right) \), using the Quotient Rule, it becomes \( \log(10^x) - \log(x(x^2+1)(x^4+2)) \).
This separation allows for easier manipulation and expansion in further steps of the process.

Power Rule

The Power Rule for logarithms allows us to simplify expressions where a term is raised to a power within a logarithm. This rule is particularly useful when the exponent is attached to the base through the logarithm, as it enables the exponent to be 'pulled out' as a multiplier. The Power Rule is expressed as \( \log(a^b) = b \cdot \log(a) \).Let's illustrate this with the example \( \log(10^x) \). According to the Power Rule, this becomes \( x \cdot \log(10) \). Using this rule makes the expressions much easier to handle and can simplify your calculations significantly. This step is crucial because it leverages one of the key properties of exponents within the context of logarithms.

Product Rule

The Product Rule provides a handy way to break down the logarithm of a product into a sum of logarithms, making it simpler to expand and solve. The rule states: \( \log(abc) = \log(a) + \log(b) + \log(c) \). When dealing with multiple factors inside a logarithm, like \( \log(x(x^2+1)(x^4+2)) \), the Product Rule allows us to treat each part separately and write it as separate logs: \( \log(x) + \log(x^2+1) + \log(x^4+2) \).
This clarity aids in further simplification and understanding of the logarithmic expression, especially when combined with other rules like the Quotient Rule and Power Rule. By breaking down larger expressions into smaller parts, this rule plays a crucial role in simplifying complex logarithmic expressions.