Problem 40 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left[\frac{100 x^{3} \sqrt[3]{5-x}}{3(x+7)^{2}}\right] $$

Step-by-Step Solution

Answer

Expanded form of the given logarithmic expression is: \[2 + 3\log(x) + \frac{1}{3}\log(5-x) - 0.477 - 2\log(x+7)\]

Step 1: Apply the Rule of Logarithm of a Quotient

Apply the logarithm of a quotient property, which states that log(a/b) = log(a) - log(b), to the provided expression: \[ \log \left[\frac{100 x^{3} \sqrt[3]{5-x}}{3(x+7)^{2}}\right] = \log [100 x^{3} \sqrt[3]{5-x}] - \log [3(x+7)^{2}] \]

Step 2: Apply the Rule of Logarithm of a Product

Apply the rule of logarithm of a product property, this states that log(a*b) = log(a) + log(b), to both parts of the expression derived from step 1: \[= \log(100) + \log(x^{3}) + \log(\sqrt[3]{5-x}) - (\log(3) + \log((x+7)^{2}))\]

Step 3: Apply the Rule of Logarithm of a Power

Apply the rule of logarithm of a power, this says, log(a^b) = b * log(a), on each part derived from step 2: \[= \log(100) + 3\log(x) + \frac{1}{3}\log(5-x) - (\log(3) + 2\log(x+7))\]

Step 4: Evaluate the logarithmic expressions where possible without a calculator

As the base of log is not mentioned, consider it as 10. So, log(100) becomes 2, and log(3) is evaluated as decimal and left as such (since without calculator it is not possible): \[= 2 + 3\log(x) + \frac{1}{3}\log(5-x) - (0.477 + 2\log(x+7))\]

Key Concepts

Logarithm of a Quotient

The logarithm function is a critical concept in mathematics, especially when dealing with division within a logarithmic expression. The property known as the logarithm of a quotient guides us through simplifying expressions like these. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator, mathematically expressed as \( \log(\frac{a}{b}) = \log(a) - \log(b) \).

When applying this to an exercise, it allows us to break down complex fractions into simpler components, making it more manageable. As seen in our example, the original complex fraction within the log can be expanded into two separate logs, one for the numerator, and one subtracted by the log of the denominator, simplifying the expression and paving the way for further expansion using other logarithm properties.

Logarithm of a Product

Multiplication within logarithmic functions brings us to the property known as the logarithm of a product. This property simplifies the process of dealing with products within a log, stating that the log of a product equals the sum of the logs of the individual factors: \( \log(a \cdot b) = \log(a) + \log(b) \).

For example, when we encounter a product within a log expression, like \( \log(100 x^{3} \sqrt[3]{5-x}) \), we can apply this property to express each factor as an individual log. This results in the sum of their logarithms, making it far less intimidating to handle. Breaking down multiplication into additive logs is beneficial because it often uncovers opportunities to simplify or evaluate parts of the expression further.

Logarithm of a Power

Dealing with powers within logarithmic functions can also be streamlined using the logarithm of a power property. This principle helps us handle expressions where the argument of the logarithm is raised to a power, articulating that \( \log(a^b) = b \cdot \log(a) \).

Such a property comes in handy when expanding logarithmic expressions. By applying it to each logarithmic part that involves an exponent, we transform multiplicative processes into additive ones, which are simpler to evaluate. In our exercise context, we convert \( \log(x^{3}) \) into \( 3\log(x) \), which clearly showcases how exponents can be taken out in front of the log, significantly untangling the original expression.