Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use the laws of logarithms to expand and simplify the expression. $$\log \frac{\sqrt{x+1}}{x^{2}+1}$$

Step-by-Step Solution

Answer

The short answer for the expanded and simplified expression is: \(\log \frac{\sqrt{x+1}}{(x^2+1)} = \frac{1}{2} \log (x+1) - \log ((x^2 + 1))\)

Step 1: Apply the quotient rule of logarithms

Using the quotient rule of logarithms (\(\log (a / b) = \log (a) - \log (b)\)), we can rewrite the given expression as: \(\log \frac{\sqrt{x+1}}{(x^2+1)} = \log (\sqrt{x+1}) - \log ((x^2 + 1))\)

Step 2: Apply the power rule of logarithms

Applying the power rule of logarithms (\(\log (a^p) = p \log (a)\)), we get: \(\log (\sqrt{x+1}) - \log ((x^2 + 1)) = \frac{1}{2} \log (x+1) - \log ((x^2 + 1))\)

Step 3: Final answer

Now we have simplified and expanded the expression: \(\log \frac{\sqrt{x+1}}{(x^2+1)} = \frac{1}{2} \log (x+1) - \log ((x^2 + 1))\)

Key Concepts

Laws of Logarithms

Logarithms are powerful mathematical tools that allow us to work with very large or small numbers more easily. The laws of logarithms provide shortcuts for simplifying expressions with logarithms. These rules are built on the fundamental properties of exponents, because logarithms themselves are essentially the inverse operations of exponents. Here are some main laws of logarithms that you should know:

• Product Rule: \( \log(a \cdot b) = \log(a) + \log(b) \). This rule shows us that the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \). This is the rule used in this exercise, allowing us to split the logarithm of a fraction into the difference of two logarithms.
• Power Rule: \( \log(a^p) = p \cdot \log(a) \). This demonstrates how to bring down exponents as coefficients in front of the logarithm.

Knowing these basic logarithmic rules allows you to efficiently tackle otherwise complex expressions, making simplification and transformation tasks much more approachable.

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential concept for transforming expressions involving fractions in logarithms. It states that the logarithm of a quotient is equal to the difference between the logarithm of the numerator and the logarithm of the denominator: \[ \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \]
This rule aligns with our understanding of division in arithmetic. If you consider how division and subtraction are related in other mathematical contexts, this rule becomes quite intuitive.
In the exercise at hand, this rule allows the expression \( \log \frac{\sqrt{x+1}}{x^2+1} \) to be broken down into two components:

• \( \log(\sqrt{x+1}) \) - the logarithm of the numerator
• \( \log(x^2+1) \) - the logarithm of the denominator

This approach simplifies complex logarithmic fractions, making further operations, like applying the power rule, possible.

Power Rule of Logarithms

The power rule of logarithms is highly useful when dealing with powers within a logarithmic argument. It states that the logarithm of a power of a number can be simplified by multiplying the logarithm of the base by the exponent:\[ \log(a^p) = p \cdot \log(a) \]
This rule gives you a means to "bring down" an exponent to the front, transforming multiplication into a simpler arithmetic form.
In our exercise, we see the power rule in action when simplifying \( \log(\sqrt{x+1}) \). The square root, \( \sqrt{x+1} \), is equivalent to \((x+1)^{1/2} \). By applying the power rule, the expression becomes:

• \( \frac{1}{2} \log(x+1) \) - illustrating how the power from the square root is now a coefficient.

This simplification technique is key not only for expanding logarithms but also for making them more manageable for further mathematical operations.