Evaluate each logarithm in three ways: (a) Use the...

Evaluate each logarithm in three ways: (a) Use the definition of logarithm to find the exact value analytically. (b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator. (c) Use a buill-in calculator finction to evaluate the logarithm. (d) Support the result of part (a) by locating the appropriate point on the graph of the finction $y=\log _{a} x$. $$\log _{16}\left(\frac{1}{8}\right)$$

Evaluate each logarithm in three ways:
(a) Use the definition of logarithm to find the exact value analytically.
(b) Support the result of part (a) by using the change-of-base rule and common logarithms on a calculator.
(c) Use a buill-in calculator finction to evaluate the logarithm.
(d) Support the result of part (a) by locating the appropriate point on the graph of the finction $y=\log _{a} x$.
$$\log _{16}\left(\frac{1}{8}\right)$$

Key Concepts

Graphical Analysis of Logarithmic Functions

Graphical analysis involves plotting the logarithmic function y = log_a x to visually inspect its behavior and locate specific points that satisfy the logarithmic equation. This method helps support the calculated results by providing an intuitive understanding of how the function increases or decreases and how it intersects specific points corresponding to the evaluated logarithm.

Calculator Functions for Logarithms

Calculator functions for logarithms offer a straightforward way to compute logarithmic values. Modern calculators come with built-in functions for common (base 10) and natural logarithms, and by applying the change-of-base rule, one can quickly determine logarithms with a different base. This method is used to verify analytically obtained results.

Change-of-Base Rule

The change-of-base rule is an important formula that allows the evaluation of logarithms with any base by relating it to logarithms in another, more convenient base, such as common logarithms (base 10) or natural logarithms (base e). The rule is given by log_a x = (log_b x) / (log_b a), and it is very practical when using calculators that only support specific logarithm functions.

Definition of a Logarithm

The definition of a logarithm is a fundamental concept that states for a given base a and a number x, the logarithm log_a x is the exponent to which a must be raised to yield x. This is expressed as a^y = x, where y is the logarithm of x with base a. This definition is used to find exact values analytically by rewriting the logarithmic equation in its exponential form.

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