Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 53

Chapter 5, Problem 53

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log₂ (x + 1)

Verified step by step guidance

Start by understanding the base function: $f(x) = \log_{2} x$. This function has a vertical asymptote at $x = 0$, a domain of $(0, \infty)$, and a range of $(-\infty, \infty)$.

Next, analyze the given function $g(x) = \log_{2} (x + 1)$. Notice that the input to the logarithm is shifted by $+1$, which means the graph of $f(x)$ is shifted horizontally to the left by 1 unit.

Determine the new vertical asymptote by setting the inside of the logarithm equal to zero: $x + 1 = 0$. Solve for $x$ to find the vertical asymptote at $x = -1$.

Use the horizontal shift to find the domain of $g(x)$. Since the logarithm requires the argument to be positive, set $x + 1 > 0$ and solve for $x$. This gives the domain as $(-1, \infty)$.

The range of $g(x)$ remains the same as the base function $f(x)$ because vertical shifts or horizontal shifts do not affect the range of a logarithmic function. Therefore, the range is $(-\infty, \infty)$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions and Their Graphs

A logarithmic function, such as f(x) = log₂(x), is the inverse of an exponential function. Its graph passes through (1,0) and increases slowly, defined only for positive x-values. Understanding the shape and behavior of the basic log function is essential for applying transformations.

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For g(x) = log₂(x + 1), the graph of log₂(x) shifts left by 1 unit. Recognizing how inside-the-function changes affect the graph helps identify new asymptotes, domain, and range.

Vertical Asymptotes and Domain of Logarithmic Functions

Logarithmic functions have vertical asymptotes where the argument equals zero, since log is undefined for non-positive values. For g(x) = log₂(x + 1), the vertical asymptote is at x = -1, which also defines the domain as all x > -1. The range remains all real numbers.

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Determining Vertical Asymptotes

Related Practice

Textbook Question

Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = (½)^x and g(x) = (½)^x-1 + 1

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. 2 log_b x + 3 log_b y

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Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₄(x+5)=3

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\ln \sqrt[3]{\frac{x}{e}}$