Problem 44 - Precalculus : Building Concepts and Connections

Question

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\frac{1}{2}\left[\log \left(x^{2}-1\right)-\log (x+1)\right]+\log x$$

Step-by-Step Solution

Answer

The simplified expression of the given problem is \(\log\left( x \sqrt{\frac{x-1}{1}}\right)\)

Step 1: Evaluate the Inner Logarithm

First, apply the quotient rule inside the square brackets in the expression. The quotient rule states that the difference of two logarithms with the same base equals the logarithm of the quotient of the two values. So, \[\frac{1}{2}\left[\log \left(x^{2}-1\right)-\log (x+1)\right] = \frac{1}{2}\left[\log \frac{x^{2}-1}{x+1}\right]\]

Step 2: Simplify the Expression Further

Continuing with the simplification, let's use the quotient rule again to simplify the expression further. This rule states that the sum of two logarithms with the same base equals the logarithm of the product of the two values. Applying this rule, we find: \[\frac{1}{2}\left[\log \frac{x^{2}-1}{x+1}\right] + \log x = \log\left( x \cdot \sqrt{\frac{x^{2}-1}{x+1}}\right)\]

Step 3: Final Reduction of the Expression

Finally, let's simplify the complex fraction inside the square root to get the final simplified logarithmic expression. Here, we are basically multiplying \(x\) with the square root of the expression \((x^{2}-1)/(x+1)\). The most simplified form of the problem is:\[\log\left( x \sqrt{\frac{x-1}{1}}\right)\]

Key Concepts

Quotient Rule in Logarithms

The quotient rule in logarithmic expressions is a fundamental concept that helps simplify differences between two logs. When you have a problem like \( \log(a) - \log(b) \), the quotient rule allows you to condense this into a single logarithm: \( \log\left(\frac{a}{b}\right) \). This is a handy way to reduce the complexity when dealing with expressions involving multiple logarithms.

In the given exercise, you noticed the expression \( \log(x^2 - 1) - \log(x + 1) \) inside the brackets. By applying the quotient rule, this simplifies to a single log: \( \log\left(\frac{x^2 - 1}{x + 1}\right) \).

This simplification step is crucial as it converts two separate logs into one, making further steps easier to execute. Always remember: the quotient rule is based on the log property that allows you to combine logs through division, greatly simplifying expressions.

Simplifying Logarithms

Simplifying logarithmic expressions can often seem puzzle-like, but by using known properties, such as the quotient rule and product rule, it becomes manageable. The term \( \frac{1}{2} \left[ \log\left(\frac{x^2 - 1}{x + 1}\right) \right] + \log x \) from the exercise is a perfect example.

First, notice the fraction \( \frac{1}{2} \), which implies a square root because \( \log(a^n) = n\log(a) \). This indicates that we are taking the square root of the term inside the log. Then, adding \( \log x \) means you can use the product rule, \( \log(a) + \log(b) = \log(a \cdot b) \), to unify the expression. This process combines the separate terms into:\[ \log\left(x \cdot \sqrt{\frac{x^2 - 1}{x + 1}}\right) \]

This meticulous step-by-step approach, simplifying each part methodically, streamlines complex problems into more manageable expressions. Consider simplifying like a mathematical detective work, uncovering simpler truths hidden under layers of complexity.

Logarithm Properties

Logarithm properties are powerful tools that underpin much of logarithmic simplification. These properties include the product rule, quotient rule, and power rule, making them key in any algebraic manipulation involving logarithms.

In our original problem, besides the quotient rule, another essential property was the application of the power rule when encountering \( \frac{1}{2}\log(x) \), which implies the square root as \( \log(\sqrt{x}) \). These properties, when correctly applied, can simplify expressions from seemingly tangled forms to their simplest versions.

• The product rule, \( \log(a) + \log(b) = \log(ab) \), helps unite multiple logarithmic terms.
• The quotient rule, \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), eliminates the separation between terms.
• The power rule, \( n\log(a) = \log(a^n) \), lets us deal with coefficients and exponents effectively.

Understanding and leveraging these properties allow you to deconstruct and rebuild expressions in a form that is often much easier to handle. Each property plays its role like a cog in a machine, collectively driving the simplification process to its logical end.