Question

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Explanation:

Analyze the given functions

Using the Logarithmic Functions knowledge point
We are given two logarithmic functions:

$$ f(x) = \log_{2}(x) \quad \text{and} \quad g(x) = \log_{5}(x) $$

Both functions have bases greater than 1 (\(2 > 1\) and \(5 > 1\)).

Evaluate the y-intercept

Using the Logarithmic Intercepts knowledge point
For any logarithmic function of the form \(y = \log_{b}(x)\), the domain is \(x > 0\). Since \(x = 0\) is not in the domain, there is no \(y\)-intercept. The first option states "Both have a y-intercept of 1", which is incorrect (they both have an \(x\)-intercept of \((1, 0)\), not a \(y\)-intercept).

Evaluate the growth behavior

Using the Logarithmic Functions knowledge point
Since both bases are greater than 1 (\(2 > 1\) and \(5 > 1\)), both functions are strictly increasing on their domain \((0, \infty)\). Therefore, both increase from left to right. This option is correct.

Evaluate the asymptotic behavior

Using the Asymptotic Behavior of Logarithms knowledge point
As \(x \to 0^+\), both \(\log_{2}(x) \to -\infty\) and \(\log_{5}(x) \to -\infty\). Thus, both graphs have a vertical asymptote at \(x = 0\) (the \(y\)-axis). This option is correct.

Evaluate the domain

Using the Logarithmic Functions knowledge point
The domain of both \(f(x) = \log_{2}(x)\) and \(g(x) = \log_{5}(x)\) is all positive real numbers, \((0, \infty)\), not all real numbers. Therefore, the option "Both have a domain of all real numbers" is incorrect.

Answer:

• Both have a y-intercept of 1
• Both increase from left to right (Correct answer)
• Both have an asymptote of x = 0 (Correct answer)
• Both have a domain of all real numbers