Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{6} 6$$.

Step-by-Step Solution

Answer

The exact value of \( \log_{6}6 \) is 1.

Step 1: Definition of a Logarithm

A logarithm is an exponent. The expression \( \log_{b}a = c \) means that \( b \) to the power of \( c \) equals \( a \). So a logarithm answers the question: 'What exponent do we need for one number to transform into another number?' It's important to apply this definition to the given logarithm.

Step 2: Applying the Definition of Logarithm

Applying the definition to \( \log_{6}6 \), the question it asks is: 'What power should 6 be raised to, to get an answer of 6?' The answer is 1, because any number (except 0) raised to the power of 1 gives the same number.

Key Concepts

Definition of Logarithm

Understanding the definition of a logarithm is a cornerstone in grasping the concept of logarithmic functions. A logarithm answers a very specific type of question: If we know a number 'a' is some power of another number 'b', what exponent 'c' was used to express 'a' in terms of 'b'? In formal terms, the logarithmic expression \( \log_{b}a = c \) expresses this relationship, telling us that \( b^c = a \).

For example, in the exercise \( \log_{6}6 \), we seek the exponent that would make the base number 6 become just 6 itself. Intuitively, we can understand that raising a number to the first power results in the number itself. Thus, \( \log_{6}6 = 1 \) because \( 6^1 = 6 \). Mastering this fundamental definition allows students to evaluate logarithms accurately without the need for a calculator.

Properties of Logarithms

Diving deeper into the world of logarithms requires familiarity with its properties, which can simplify the process of solving logarithmic equations and make calculations more manageable. Some essential properties include:

• Product Rule: \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \), which explains that the logarithm of a product is the sum of the logarithms.
• Quotient Rule: \( \log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) - \log_{b}(n) \), indicating that the logarithm of a quotient is the difference between the logarithms.
• Power Rule: \( \log_{b}(m^k) = k \times \log_{b}(m) \), which allows us to move the exponent outside the logarithm as a multiplier.
• Change of Base Formula: \( \log_{b}(m) = \frac{\log_{k}(m)}{\log_{k}(b)} \), valuable for converting logarithms to a different base.

These rules empower students to tackle complex logarithmic expressions step by step, decomposing them into more elementary parts that can be easier to interpret and solve.

Exponential Functions

Exponential functions represent the inverse of logarithmic functions and are instrumental in modeling growth and decay processes in various disciplines like biology, economics, and physics. An exponential function is typically of the form \( f(x) = a \times b^x \), where 'a' represents the initial value, 'b' is a positive base other than 1, and 'x' relates to the exponent. This structure encapsulates the concept of consistent multiplicative change; as 'x' increases by 1, the function's value is multiplied by 'b'.

For example, considering the exponential function \( 6^x \), the growth of the function is determined by the value assigned to 'x'. If we were to solve for the instance when \( 6^x = 6 \), by understanding exponential functions, we'd know that \( x = 1 \), precisely because any base raised to the first power returns the base. Equally, the inverse process, which would involve the logarithm \( \log_{6}6 \), reaffirms this relationship. Comprehending exponential functions equips learners with insights into not only abstract mathematical concepts but also real-world phenomena where exponential change occurs.