Problem 1 - Algebra and Trigonometry

Question

Evaluate the expression. $$ \log _{3} \sqrt{27} $$

Step-by-Step Solution

Answer

\( \frac{3}{2} \)

Step 1: Convert Square Root to Exponentiation

The square root of a number can be expressed as raising that number to the power of 1/2. Therefore, rewrite \( \sqrt{27} \) as \( 27^{1/2} \).

Step 2: Apply the Logarithmic Property of Powers

One of the logarithmic identities states that \( \log_b{(a^c)} = c \log_b{a} \). Applying this to our expression, we get: \[ \log_3{(27^{1/2})} = \frac{1}{2} \log_3{27}. \]

Step 3: Simplify \( \log_3{27} \)

Note that 27 can be expressed as \( 3^3 \). Therefore, substitute \( 3^3 \) for 27: \[ \log_3{27} = \log_3{(3^3)}. \] Using the identity \( \log_b{(b^c)} = c \), we have \( \log_3{(3^3)} = 3 \).

Step 4: Compute the Final Expression

Substitute the result from the previous step into the expression from Step 2: \[ \frac{1}{2} \log_3{27} = \frac{1}{2} \times 3 = \frac{3}{2}. \]

Key Concepts

Logarithmic Identities

When working with logarithms, it's important to understand their identities. These identities simplify complex expressions and equations. One popular identity is the power rule: if you have a logarithm of a power, like \( \log_b{(a^c)} \), you can rewrite it as \( c \cdot \log_b{a} \). This means that the exponent \( c \) can be taken out of the logarithm. This identity is especially useful for expressions involving exponents, making them easier to handle or evaluate in equations.

There are also other key identities that are useful, such as:

• The product rule: \( \log_b{(xy)} = \log_b{x} + \log_b{y} \)
• The quotient rule: \( \log_b{(\frac{x}{y})} = \log_b{x} - \log_b{y} \)
• The change of base formula: \( \log_b{a} = \frac{\log_c{a}}{\log_c{b}} \)

These rules help in transforming and comparing expressions, making logarithmic calculations far more manageable across different scientific and engineering tasks.

Properties of Logarithms

Logarithms possess properties that provide multiple ways to simplify and understand logarithmic expressions. One such significant property is that a logarithm with the same base for both its argument and base simplifies directly to the exponent. In formal terms, for \( \log_b{(b^c)} \), the result is the exponent \( c \). For example, \( \log_3{(3^3)} \) simplifies to 3: you directly pull down the exponent because of this property.

Besides this, logarithms act inversely to exponentiation, meaning they can "undo" exponentiation operations, allowing extraction of roots or solving exponential functions. This inverse relation is why, for instance, the logarithm of a square root like \( \sqrt{27} \) first involves expressing it as \( 27^{1/2} \), then applying the power rule to simplify.

Understanding these properties clearly lays a foundational step in mathematics necessary for complex calculations and helps in developing problem-solving strategies in algebra, calculus, and beyond.

Exponentiation

Exponentiation refers to the operation of raising a number to a power. This operation is indicated by a superscript number. For instance, in \( 27^{1/2} \), 27 is the base and \( 1/2 \) is the exponent, indicating a square root in this case.

Key things to remember with exponentiation include:

• An exponent of 1/2 represents a square root, so \( a^{1/2} \) is the same as \( \sqrt{a} \).
• An exponent of 1 represents the number itself \( a^1 = a \).
• An exponent of zero \( a^0 \) equals 1 for any non-zero number \( a \).

These rules simplify interpreting or restructuring mathematical expressions. It's this flexibility and ease of transforming or manipulating expressions through exponent manipulation that make exponentiation a cornerstone concept in math.

The use of exponentiation is extensive, crossing various branches of math and impacting fields such as computer science, physics, and engineering, where exponents are used to express large numbers or quantities efficiently.