Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression without using a calculator.\(\log _{6} \frac{1}{36}\)

Step-by-Step Solution

Answer

-2

Step 1: Identify the Base and Argument

In the logarithmic expression \(\log _6\frac{1}{36}\), 6 is the base and \(\frac{1}{36}\) is the argument. Here, recognize that \(\frac{1}{36}\) is the same as \(6^{-2}\).

Step 2: Apply Logarithm Property

Now, recall the logarithmic property that \(\log_b (b^x)=x\), where b is the base and x is the exponent. Utilizing this property, the expression \(\log _6 {6^{-2}}\) becomes simply \(-2\).

Step 3: State the Answer

Therefore, \(\log _6\frac{1}{36}\) equals -2.

Key Concepts

Base and Argument

When dealing with logarithms, two main components come into play: the base and the argument. The base is the number that logarithms use as a reference for scaling. For instance, in a logarithmic expression like \( \log_6 \frac{1}{36} \), the number 6 is the base.

The argument is the value for which you are finding the logarithm. In this case, \( \frac{1}{36} \) is the argument. Understanding these two components is crucial because it helps determine the scale and position of the argument relative to the base.

In essence, asking for \( \log_b a \) is tantamount to asking: "To what power must we raise the base \( b \) in order to achieve the argument \( a \)?" By breaking it down like this, you can better visualize and understand the role each part plays in the expression.

Logarithm Properties

Logarithms come with a set of handy properties that make calculations and simplifications a whole lot easier. One critical property to remember is the power rule: \( \log_b (b^x) = x \). This states that if the argument \( a \) is a power of the base \( b \), then the logarithm simplifies directly to the exponent \( x \).

For example, in the expression \( \log_6 6^{-2} \), we notice that the argument is \( 6^{-2} \), which is a power of the base 6. Hence, by applying this logarithmic property, it directly simplifies to the result \(-2\).

This property is invaluable for simplifying complex expressions where the argument can be expressed as a power of the base. Recognizing and applying these properties fluently can greatly ease the process of solving logarithmic problems.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. They are closely related to logarithms, as logarithms essentially invert exponential operations.

In the expression \( \log_6 \frac{1}{36} \), it is helpful to recognize \( \frac{1}{36} \) as an exponent of the base 6: \( 6^{-2} \). This representation shows that \( 6 \) must be raised to the power of \(-2\) to yield \( \frac{1}{36} \).

A key understanding of exponents is their role in expressing the inverse operations with logarithms. For instance, if \( y = b^x \), then \( \log_b y = x \). Envisioning exponents and their inverses not only aids in quickly evaluating expressions but also build a deeper grasp of their interplay with logarithms.