The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = ln(x/2)

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Start with the base function $f(x) = \ln x$, which has a vertical asymptote at $x = 0$, domain $(0, \infty)$, and range $(-\infty, \infty)$.

Recognize that $h(x) = \ln\left(\frac{x}{2}\right)$ can be rewritten using logarithm properties as $h(x) = \ln x - \ln 2$, which represents a horizontal shift of the graph of $\ln x$.

Since $\ln\left(\frac{x}{2}\right)$ is defined only when $\frac{x}{2} > 0$, determine the domain of $h(x)$ as $x > 0$.

Identify the vertical asymptote of $h(x)$ by setting the inside of the logarithm equal to zero: $\frac{x}{2} = 0$ implies $x = 0$, so the vertical asymptote remains at $x = 0$.

Note that the range of $h(x)$ is the same as the base function $\ln x$, which is $(-\infty, \infty)$, because logarithmic transformations involving horizontal shifts do not affect the range.

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Key Concepts

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

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the inverse of the exponential function e^x. It is defined only for positive x-values and has a vertical asymptote at x = 0. Understanding its basic shape and properties is essential for graphing transformations.

Function Transformations

Transformations involve shifting, stretching, compressing, or reflecting the graph of a function. For h(x) = ln(x/2), the input is scaled horizontally, which shifts the graph and affects the location of asymptotes. Recognizing how changes inside the function argument affect the graph is key.

Domain, Range, and Asymptotes

The domain of ln(x) is (0, ∞), and its range is all real numbers. Transformations can change the domain by shifting the vertical asymptote, which is where the function is undefined. Identifying asymptotes helps determine domain restrictions and understand the graph's behavior.

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In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $8^{x} = 12143$

Textbook Question

The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.

h(x) = ln (2x)

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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. log₅ x+log₅(4x−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. $$\log$ x + $\log$(x^2 - 1) - $\log$ 7 - $\log$(x + 1)$

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