Problem 53 - Elementary and Intermediate Algebra

Question

Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\log \sqrt{5}\)

Step-by-Step Solution

Answer

\(\log \sqrt{5} = \frac{1}{2} \log 5\)

Step 1: Express the Radical as an Exponent

The square root operation can be represented as an exponent of 1/2. Thus, \( \sqrt{5} \) can be rewritten as \( 5^{1/2} \).

Step 2: Apply Logarithmic Power Rule

The logarithmic power rule states that \( \log(a^b) = b \cdot \log(a) \). Apply this to \( \log(5^{1/2}) \), which gives \( \frac{1}{2} \cdot \log(5) \).

Step 3: Write the Final Expression

The expression for \( \log \sqrt{5} \) without using an exponent or a radical is \( \frac{1}{2} \log 5 \).

Key Concepts

Logarithmic Power Rule

The logarithmic power rule is a handy tool when dealing with exponents inside a logarithm. It simplifies expressions of the form \( \log(a^b) \) to \( b \cdot \log(a) \). This means you can "bring down" the exponent as a multiplier in front of the logarithm. For example, in our exercise, \( \log(5^{1/2}) \), the exponent \( 1/2 \) becomes a coefficient: \( \frac{1}{2} \cdot \log(5) \).
This rule is particularly useful for making calculations easier or when attempting to express logarithms in simpler or more manageable terms. By reducing exponential forms, we often prepare the expression for further manipulation or simplification. It is essential to remember the base of the logarithm should remain unchanged while only the exponent is handled.

Radical Expressions

Radical expressions often involve roots, such as square roots or cube roots. These can sometimes be tricky to work with, but they can also be rewritten in an alternative form using exponents.
For instance, the square root of a number, say \( \sqrt{5} \), can be expressed as \( 5^{1/2} \). This transformation is beneficial when you need to apply algebraic rules such as the logarithmic power rule.

• Convert square roots (or other roots) into exponential form to simplify operations.
• Understand that a square root is equivalent to raising a number to the power of \( 1/2 \).

By changing the form of a radical into an exponent, you open up possibilities to use a broader range of algebraic strategies, including using logarithmic rules. This concept allows for tackling more complex problems with relative ease.

Simplifying Logarithms

Simplifying logarithms makes them easier to understand and work with, especially when dealing with advanced problems or further calculations. The essence of simplifying a logarithm is transforming it into a form without complex symbols like radicals or high exponents.

• Start by expressing any radical symbols as exponents.
• Apply the logarithmic power rule to bring down exponents.
• Further reduce the expression by combining or breaking down logs if necessary.

Consider our example: \( \log \sqrt{5} \) was initially in an intricate root form. By rewriting it as \( \log(5^{1/2}) \) and applying the power rule, we ended up with \( \frac{1}{2} \log 5 \). The absence of a radical or exponent in the logarithm simplifies it significantly, making it ready for substitution or evaluation if required. Simplifying not only enhances comprehension but also eases the computational load in extensive expressions.