Let’s work through the problem step by step:

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Step 1: Base graph $f(x) = \log_4(x)$:

The base graph $f(x) = \log_4(x)$ has the following properties:

• Vertical asymptote: $x = 0$, because $\log_4(x)$ is undefined for $x \leq 0$.
• Domain: $(0, \infty)$.
• Range: $(-\infty, \infty)$.

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Step 2: Transformations to graph $g(x) = \log_4(x + 6)$:

The function $g(x) = \log_4(x + 6)$ introduces a horizontal shift of $6$ units to the left (since $x + 6 = 0$ when $x = -6$).

Properties of $g(x)$:

• Vertical asymptote: The vertical asymptote shifts from $x = 0$ to $x = -6$.
• Domain: $x + 6 > 0$, so $x > -6$. The domain is $(-6, \infty)$.
• Range: Logarithmic functions always have a range of $(-\infty, \infty)$, which remains unchanged.

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Graphical Transformation:

The graph of $g(x)$ is obtained by shifting the graph of $f(x) = \log_4(x)$ 6 units to the left.

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Step 3: Vertical Asymptote and Range:

• Vertical Asymptote: $x = -6$.
• Domain: $(-6, \infty)$.
• Range: $(-\infty, \infty)$.

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Visualization:

I will now generate and share a graph of $f(x) = \log_4(x)$ and $g(x) = \log_4(x + 6)$. Define the transformed function g(x) = log_4(x + 6) def log_base_4_transformed(x): return np.log(x + 6) / np.log(4)

Compute the transformed function

g_x = log_base_4_transformed(x)

Plot the graphs

plt.figure(figsize=(10, 6)) plt.plot(x, f_x, label=r"$f(x) = \log_4(x)$", color='blue', linewidth=2) plt.plot(x, g_x, label=r"$g(x) = \log_4(x + 6)$", color='green', linestyle='--', linewidth=2)

Highlight the vertical asymptotes

plt.axvline(0, color='black', linestyle=':', linewidth=1, label="As