Solve each problem. Sales (in thousands of units) of a new...

Solve each problem. Sales (in thousands of units) of a new product are approximated by the logarithmic function $$ S(t)=100+30 \log _{3}(2 t+1) $$ where $t$ is the number of years after the product is introduced. (a) What were the sales, to the nearest unit, after 1 yr? (b) What were the sales, to the nearest unit, after 13 yr? (c) Graph $y=S(t)$

Solve each problem.
Sales (in thousands of units) of a new product are approximated by the logarithmic function
$$
S(t)=100+30 \log _{3}(2 t+1)
$$
where $t$ is the number of years after the product is introduced.
(a) What were the sales, to the nearest unit, after 1 yr?
(b) What were the sales, to the nearest unit, after 13 yr?
(c) Graph $y=S(t)$

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and typically have the form f(x) = a + b log_c(x) where c is the base of the logarithm. They are characterized by slow growth for large input values and have a vertical asymptote that represents the limit of the domain. These functions are useful in modeling processes that initially show rapid change but then level off.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to compute its output. This is a fundamental technique in algebra and calculus used to determine specific outcomes of a model or function at given input values.

Transformation of Functions

Transformations involve modifying a basic function by shifting, stretching, or compressing it. In the given function, the parameters alter the shape and position of the logarithmic curve: a vertical shift (the constant term), a vertical stretch (the multiplier of the logarithm), and a horizontal scaling (inside the logarithm argument) are all examples of how a base graph is transformed.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all positive values of its argument because the logarithm is undefined for zero or negative numbers. Identifying the domain is crucial when working with logarithmic functions to ensure that the inputs are within a valid range.

Graphical Representation

Graphical representation entails plotting a function to visually capture its behavior, such as intercepts, growth patterns, and asymptotes. For transformed logarithmic functions, recognizing how shifts, stretches, and compressions affect the basic log graph is key to accurately sketching the graph.

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