Question

Write the expression as the logarithm of a single quantity. $$ \frac{3}{2}\left[\ln \left(x^{2}+1\right)-\ln (x+1)-\ln (x-1)\right] $$

Short answer

The single logarithm expression of the original function is \( \ln \left ( \left ( \frac{x^{2}+1}{x^2 - 1} \right)^{\frac{3}{2}} \right )\)

Step-by-step solution

Simplify the Logarithm

Use the logarithmic property \(\ln(a)-\ln(b) = \ln(\frac{a}{b})\). Applying this to the expression gives: \[ \frac{3}{2} \ln \left ( \frac{x^{2}+1}{(x+1)(x-1)} \right )\]

Apply Power Law of Logarithms

Use the power rule of logarithms, which states that \(\ln a^n = n \ln a\). This allows to move the \(\frac{3}{2}\) in front of the logarithm inside as a power. So, the expression becomes: \[ \ln \left ( \left ( \frac{x^{2}+1}{(x+1)(x-1)} \right)^{\frac{3}{2}} \right )\]

Simplify the Fraction inside the Logarithm

Try to simplify the fraction inside the logarithm: \[ \ln \left ( \left ( \frac{x^{2}+1}{x^2 - 1} \right)^{\frac{3}{2}} \right )\]

Key concepts

Logarithmic Properties

Understanding the properties of logarithms is like having a Swiss Army knife for solving algebraic puzzles. These properties aren't just math rules; they're tools that can simplify complex expressions and reveal the hidden relationships between numbers. One of the fundamental properties is the quotient rule, which states that the logarithm of a division between two values is equal to the difference of their logarithms, formally written as \( \log_b(\frac{a}{b}) = \log_b(a) - \log_b(b) \). When you encounter a hefty expression like \( \frac{3}{2}[\ln(x^2+1)-\ln(x+1)-\ln(x-1)] \), knowing this rule allows you to combine the logarithms into one statement, unraveling the problem and bringing you a step closer to the solution.

Another key feature to recall is the product rule for logarithms, \( \log_b(a \cdot c) = \log_b(a) + \log_b(c) \), which is the reverse mechanism of the quotient rule and is applied when multiplying quantities within the logarithmic function. Recognizing when and how to apply these logarithmic properties will foster not just computational ease but also a deeper comprehension of how logarithmic functions operate and interact.

Power Rule of Logarithms

Turn up the power with the power rule of logarithms! This property is the secret sauce to streamlining expressions with exponents. It tells us that the logarithm of a power, like \( a^n \), is the same as the exponent times the logarithm of the base: \( \log_b(a^n) = n \times \log_b(a) \). This rule comes particularly handy when you're trying to condense an expression that has a coefficient in front of a logarithm, such as \( \frac{3}{2}\ln(x^2+1) \).

By applying the power rule, you can effectively 'fold' the coefficient into the logarithm as an exponent, simplifying the expression to \( \ln((x^2+1)^{\frac{3}{2}}) \) and eliminating the need for that initially pesky fraction in front. Not just a cosmetic change, moving the coefficient inside the logarithm using the power rule often paves the way to further simplification and can reveal how the parts of an expression depend on each other.

Simplifying Expressions

When facing the wilderness of algebra, simplifying expressions is like finding a clear path through an overgrown forest. It's the act of making complex, cluttered mathematical statements neat and approachable. Simply put, it's turning the math 'mess' into a 'success'.

Simplifying can involve a variety of techniques, such as combining like terms, factoring polynomials, or reducing fractions. Take the expression \( \ln\left(\left(\frac{x^2+1}{x^2-1}\right)^{\frac{3}{2}}\right) \); at this stage, our initial complex logarithmic expression has been tamed. But you're not out of the mathematical woods yet! Examining the fraction inside more closely reveals it is ready for further action—in this case, recognizing that \(x^2 - 1\) is a difference of squares that can be factored to \((x+1)(x-1)\).

Such insights can only be reached through a deep familiarity with algebraic manipulation, turning intimidating equations into friendly, solvable puzzles. At each stage, double-check your work, as every step towards simplification is a step towards unveiling the numerical truth hidden beneath the surface.