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The change-of-base formula is a useful tool when evaluating logarithms, especially when working with bases other than 10 or when a given calculator does not support a specific base logarithm. The formula is given as:

• \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \)

This equation allows you to convert a logarithm with any base \( b \) to a logarithm of a different base \( c \), which is commonly base 10 (\( \log \)) or base \( e \) (\( \ln \)), since these bases are available on most calculators.

In the given exercise, the change-of-base formula was used to transform \( \log_{16} \left( \frac{1}{8} \right) \) into a form that can easily be calculated using a typical calculator. We found that \( \frac{\log \left( \frac{1}{8} \right)}{\log(16)} \) yields \( -0.75 \), confirming the analytical result arrived at via direct calculation.

Graphing provides a visual method to confirm logarithmic values. A logarithmic function of the form \( y = \log_a(x) \) can be plotted on a graph, where \( y \) represents the logarithm and \( x \) is the value of which the logarithm is taken.

When graphing \( y = \log_{16}(x) \), each point on the graph has coordinates \( (x, y) \), where \( y = \log_{16}(x) \). For the exercise in question, locating \( x = \frac{1}{8} \) on this graph, the corresponding \( y \)-value should be \( -\frac{3}{4} \). This confirms that \( \log_{16} \left( \frac{1}{8} \right) = -\frac{3}{4} \). Such graphical interpretation not only verifies numerical answers but helps illustrate how logarithms behave with different bases and arguments.

The concept of exponents is foundational to understanding logarithms, as a logarithm essentially asks the question: "To what power must the base \( b \) be raised, to yield the number \( a \)?" Therefore, the relationship between exponents and logarithms is interchangeable, described by the identity:

• If \( b^c = a \), then \( \log_b(a) = c \).

To tackle logarithmic problems, you might need to manipulate or rewrite numbers as their exponential forms. In our exercise, the numbers \( 16 \) and \( \frac{1}{8} \) were rewritten as \( 2^4 \) and \( 2^{-3} \) respectively, to use identical bases. Simplifying \( (2^4)^c = 2^{-3} \), we equate exponents to solve for \( c \) given \( 4c = -3 \), resulting in \( c = -\frac{3}{4} \). Mastering exponents is critical for effectively navigating and confirming outcomes in logarithmic calculations.