Use properties of logarithms to rewrite each function, and describe how the graph of the given function compares to the graph of g(x) = ln x. ƒ(x) = ln(e²x)

Verified step by step guidance

1. Start by recognizing the given function: \(\displaystyle f(x) = \ln(e^{2}x)\). Notice that the argument of the logarithm is a product of \(e^{2}\) and \(x\).
2. Use the logarithm property that states \(\ln(ab) = \ln a + \ln b\) to separate the logarithm of the product: \(\ln(e^{2}x) = \ln(e^{2}) + \ln(x)\).
3. Recall that \(\ln(e^{k}) = k\) for any constant \(k\), so simplify \(\ln(e^{2})\) to \$2\(. This gives \)f(x) = 2 + \ln(x)$.
4. Interpret the transformation: since \(f(x) = \ln(x) + 2\), this represents a vertical shift of the graph of \(g(x) = \ln x\) upward by 2 units.
5. Summarize the effect on the graph: the shape of the graph remains the same as \(g(x) = \ln x\), but every point is moved 2 units higher on the y-axis.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. For example, ln(a^b) = b ln(a), and ln(ab) = ln(a) + ln(b). These properties allow us to simplify or rewrite logarithmic expressions to make them easier to analyze or graph.

Natural Logarithm Function g(x) = ln(x)

The natural logarithm function ln(x) is the inverse of the exponential function e^x. Its graph passes through (1,0), is defined for x > 0, and increases slowly. Understanding its shape and domain is essential for comparing transformations of logarithmic functions.

Function Transformations and Graph Comparisons

Transformations such as shifts, stretches, and compressions affect the graph of a function. When rewriting ƒ(x) = ln(e^{2x}), recognizing it simplifies to a linear transformation of ln(x) helps describe how its graph compares to g(x) = ln(x), such as horizontal scaling or vertical shifts.