Use a graphing utility and the change-of-base property to graph each function. y = log₃ x

Verified step by step guidance

1. Understand that the function given is \( y = \log_3 x \), which is a logarithmic function with base 3. This means it gives the exponent to which 3 must be raised to get \( x \).
2. Recall the change-of-base formula for logarithms: \( \log_a b = \frac{\log_c b}{\log_c a} \), where \( c \) is any positive number different from 1. This allows us to rewrite \( \log_3 x \) in terms of common logarithms (base 10) or natural logarithms (base \( e \)).
3. Apply the change-of-base formula to rewrite \( y = \log_3 x \) as \( y = \frac{\log x}{\log 3} \) if using common logarithms, or \( y = \frac{\ln x}{\ln 3} \) if using natural logarithms.
4. Use a graphing utility (such as a graphing calculator or software) to plot the function \( y = \frac{\log x}{\log 3} \) or \( y = \frac{\ln x}{\ln 3} \). This will give the graph of \( y = \log_3 x \).
5. Analyze the graph: note that the domain is \( x > 0 \), the graph passes through the point \( (1,0) \) because \( \log_3 1 = 0 \), and the graph increases slowly as \( x \) increases.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function and is written as y = log_b(x), where b is the base. It answers the question: to what power must the base b be raised to produce x? Understanding the domain (x > 0) and range (all real numbers) is essential for graphing.

Change-of-Base Formula

The change-of-base formula allows you to rewrite logarithms with any base b as a ratio of logarithms with a different base, typically base 10 or e: log_b(x) = log_c(x) / log_c(b). This is useful when graphing with calculators or utilities that only support common or natural logs.

Using Graphing Utilities

Graphing utilities like calculators or software can plot functions by evaluating points. Since many utilities do not support arbitrary log bases directly, applying the change-of-base formula lets you graph y = log_3(x) by inputting y = log(x)/log(3), enabling visualization of the function's shape and behavior.