Problem 40 - College Algebra

Question

In Exercises \(1-40,\) 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

The expanded form of the given expression by using properties of logarithms is: \(2 + 3\log[x] + \frac{1}{3}\log[5-x] - \log[3] - 2\log[x+7]\)

Step 1: Apply the logarithm quotient rule

Use the rule \(\log_b(m/n) = \log_b(m) - \log_b(n)\) to separate the numerator and denominator of the fraction inside the log. So, it will become: \(\log[100 x^{3} \sqrt[3]{5-x}] - \log[3(x+7)^{2}]\)

Step 2: Apply the logarithm product rule

Now use the product rule \(\log_b(mn) = \log_b(m) + \log_b(n)\) to separate into multiple logs the expression \(\log[100 x^{3} \sqrt[3]{5-x}]\). It will become: \(\log[100] + \log[x^{3}] + \log[\sqrt[3]{5-x}]\)

Step 3: Apply the logarithm power rule and evaluate

Apply the power rule \(\log_b(m^n) = n\log_b(m)\) for the expressions \(\log[x^{3}]\) and \(\log[\sqrt[3]{5-x}]\), and evaluate \(\log[100]\). Thus it becomes: \(2 + 3\log[x] + \frac{1}{3}\log[5-x]\)

Step 4: Apply the logarithm product and power rule to the second term

Apply the product rule to \(\log[3(x+7)^{2}]\) as \(\log[3] + \log[(x+7)^{2}]\) and then apply the power rule to \(\log[(x+7)^{2}]\). This becomes: \(\log[3] + 2\log[x+7]\)

Step 5: Combine all logs

Now we combine all the logarithmic expressions from the previous steps. Thus, the completely expanded form becomes: \(2 + 3\log[x] + \frac{1}{3}\log[5-x] - \log[3] - 2\log[x+7]\)

Key Concepts

Logarithm Quotient Rule

The logarithm quotient rule is an essential property when dealing with logarithmic expressions containing fractions. This rule states that the logarithm of a quotient is the difference of the logarithms:

• \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)

To visualize this, imagine you have a log expression like \( \log_b(a/b) \). Using the quotient rule, you can split it into two separate log expressions: \( \log_b(a) - \log_b(b) \). This helps break down complex expressions into simpler parts, making them easier to manage and understand.
In our original exercise, we started by applying the quotient rule to break up the fraction in the log expression. This allowed us to manage the numerator and the denominator separately, like peeling layers off an onion. This simplification is the first step toward solving complex logarithmic expressions.

Logarithm Product Rule

The logarithm product rule comes into play when dealing with products inside a logarithm. The rule allows us to separate a log of a product into the sum of two separate logarithms:

• \( \log_b(mn) = \log_b(m) + \log_b(n) \)

Picture a log expression such as \( \log_b(xy) \). By applying the product rule, you can expand this into \( \log_b(x) + \log_b(y) \). This process is akin to unpacking a box, where each item inside is dealt with individually rather than all at once.
In our exercise, the product rule was used to further break down the expanded numerator log expression. We applied it to \( \log[100 x^3 \sqrt[3]{5-x}] \), allowing us to manage each component separately, ultimately simplifying a challenging problem into more digestible pieces.

Logarithm Power Rule

The logarithm power rule is a cornerstone property, especially convenient when dealing with powers within logs. This rule states that the logarithm of a number with an exponent can be rewritten by moving the exponent in front of the log:

• \( \log_b(m^n) = n \log_b(m) \)

Think of a power inside a logarithmic expression like \( \log_b(x^n) \). By applying the power rule, this turns into \( n \log_b(x) \). Like simplifying a tall stack of blocks by removing layers, this rule makes manipulating and understanding complex logs much easier.
In our example problem, the power rule helped us handle terms like \( x^3 \) and \( (5-x)^{1/3} \) efficiently. By pulling down the exponents in each term, these were converted into straightforward multiplications, leading us closer to fully resolving the logarithmic expression after combining with other properties.