Problem 30 - Algebra A Combined Function

Question

Find the exact value of each logarithm. $$ \log \sqrt{10} $$

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Understanding the Problem

We need to find the exact value of the logarithm \( \log \sqrt{10} \). This means finding the power to which the base (implicitly 10 when not specified) must be raised to get \( \sqrt{10} \).

Step 2: Rewriting the Expression

The expression \( \sqrt{10} \) can be rewritten using fractional exponents. Thus, \( \sqrt{10} = 10^{1/2} \).

Step 3: Using Logarithm Power Rule

Utilize the power rule of logarithms: \( \log(a^b) = b \cdot \log(a) \). Applying it here:\[ \log(10^{1/2}) = \frac{1}{2} \cdot \log(10) \]

Step 4: Simplifying Using Logarithm Identity

We know that \( \log(10) = 1 \) because 10 is the base of the common logarithms (base-10). Substitute this value into the expression:\[ \frac{1}{2} \cdot 1 = \frac{1}{2} \]

Step 5: Final Answer

Thus, the exact value of \( \log \sqrt{10} \) is \( \frac{1}{2} \).

Key Concepts

Logarithm Power Rule

The logarithm power rule is a very useful tool when solving logarithmic expressions. It states: \( \log(a^b) = b \cdot \log(a) \). This rule helps simplify logarithms that contain exponents. By using this rule, you can pull the exponent in the logarithmic function out as a multiple.
This process makes calculations easier and more straightforward.

• If you have a logarithm, such as \( \log(10^{1/2}) \), you can apply the power rule to simplify it to \( \frac{1}{2} \cdot \log(10) \).
• This extraction makes it clear how exponents manipulate logarithmic values.

Understanding and applying the power rule is essential for efficiently solving many logarithmic problems. It allows you to quickly break down more complex expressions into simpler, solvable parts.

Fractional Exponents

Fractional exponents are a way of expressing roots in a different form. Instead of writing roots like \( \sqrt{10} \), you can express the same operation with exponents: \( 10^{1/2} \). This notation helps in simplifying calculations, especially when dealing with logarithms.

• For example, the square root of a number, \( \sqrt{x} \), can be rewritten as \( x^{1/2} \).
• Likewise, a cube root, or \( \sqrt[3]{x} \), becomes \( x^{1/3} \).

Using fractional exponents in logarithms allows you to apply rules like the power rule, simplifying the problem to a more manageable form. This is particularly beneficial in algebra and calculus, where such manipulations are common.

Common Logarithms

Common logarithms are logarithms with base 10. They are frequently used in mathematics, science, and engineering because our number system is base 10. When you see a logarithm without an explicitly written base, it's usually a common logarithm. The notation "log" without a subscript implies base 10.

• For instance, \( \log(100) \) asks "to what power must 10 be raised to result in 100?" The answer is \( 2 \), since \( 10^2 = 100 \).
• Common logarithms make calculations intuitive when working with powers of 10 since humans commonly use these numbers.

Understanding common logarithms simplifies learning concepts about scales like decibels in acoustics or Richter scale for earthquakes, where powers of ten come into play.

Logarithm Identity

A crucial identity in logarithms is that \( \log_{10}(10) = 1 \). This identity states that any number raised to the power of its own base gives 1 when expressed in logarithms. In the context of common logarithms: \( \log(10) = 1 \).
Such an identity is a foundation for simplifying more complex expressions.

• For example, if you have an expression like \( \frac{1}{2} \cdot \log(10) \), knowing \( \log(10) = 1 \) allows you to quickly simplify it to \( \frac{1}{2} \).
• This helps in breaking down and verifying results efficiently.

This identity is often used implicitly in mathematical problems involving logarithms, making it a handy knowledge piece when working through textbook exercises or real-world applications.