Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \sqrt{a-1}, \quad a>1\)

Step-by-Step Solution

Answer

The expanded form of the expression using the properties of logarithms is \(\frac{1}{2} \ln (a-1)\).

Step 1: Express Square Root as Power of 0.5

Rewrite the square root in the expression as a power of \(\frac{1}{2}\). Therefore, the expression \(\ln \sqrt{a-1}\) can be written as \(\ln (a-1)^{\frac{1}{2}}\).

Step 2: Application of Logarithm Power Rule

The power rule states that \(\log_b (m^n) = n \log_b m\). Hence, \(\ln (a-1)^{\frac{1}{2}}\) can be rewritten as \(\frac{1}{2} \ln (a-1)\).

Step 3: Final Simplified Expression

After applying the properties as described, the final simplified expression is \(\frac{1}{2} \ln (a-1)\)

Key Concepts

Logarithm Expansion

When you first encounter a logarithmic function, it can seem a bit like a secret code. However, there are certain properties and rules, like the expansion of logarithms, that make these expressions more manageable. Logarithm expansion is the process of breaking down a logarithm into simpler parts, using specific properties to express it as a sum, difference, or product of logarithms.
In our example, we start with the expression \(\ln \sqrt{a-1}\). The key here is to notice that the square root can be expressed as a power, specifically \(a-1\) raised to the power of 0.5. Understanding this allows us to rewrite the logarithm in a different, often simplified form.

• Converting roots to fractional powers.
• Expressing logarithms based on these power rules.

By converting \(\sqrt{a-1}\) into \((a-1)^{\frac{1}{2}}\), the expression became more accessible for further manipulation.

Power Rule

The Power Rule is a fundamental tool in logarithms that allows us to move exponents out in front of the logarithm. The Power Rule states \(\log_b (m^n) = n \log_b m\).
This is handy when you wish to simplify expressions that involve powers, as it trades a power for a coefficient, which is often easier to deal with in calculations.
In the example \(\ln (a-1)^{\frac{1}{2}}\), the Power Rule is applied by moving the exponent \(\frac{1}{2}\) in front of the logarithm:
\[\ln (a-1)^{\frac{1}{2}} = \frac{1}{2} \ln (a-1)\]

• Exponent becomes a coefficient.
• Simplifies manipulation of logarithms.

This transformation makes calculations more straightforward and sets you up for smooth subsequent steps in problem-solving.

Simplification of Logarithmic Expressions

After applying logarithmic identities, simplification is often the ultimate goal. Simplifying a logarithmic expression means rewriting it in its most compact and straightforward form, which often reveals more about the relationships within an expression.
In our specific example, once \(\ln (a-1)^{\frac{1}{2}}\) is transformed using the Power Rule, we end with \(\frac{1}{2} \ln (a-1)\).

• Final expression is simpler and cleaner.
• Reveals the basic structure and relations.

This type of simplification is especially useful in solving equations, integrating calculus problems, or when logarithmic expressions need to be combined or compared.