Question

in the problems below, \\(f(x) = \log_{2}x\\) and \\(g(x) = \log_{10}x\\).

how are the graphs of \\(f\\) and \\(g\\) similar? check all that apply.

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

which point do the graphs of \\(f\\) and \\(g\\) have in common?

for \\(x > 1\\), the graph of which function increases faster?

Explanation:

Analyze the given logarithmic functions

Using the Logarithmic Functions knowledge point

$$
LATEXBLOCK0
$$

Determine the common point

Using the Logarithmic Intercepts knowledge point

$$
LATEXBLOCK1
$$

Compare growth rates for \(x > 1\)

We evaluate the functions at a value \(x > 1\), for example \(x = 10\):

$$
LATEXBLOCK2
$$

Since the base \(2 < 10\), the function with the smaller base, \(f(x) = \log_{2}x\), increases faster for \(x > 1\).

Answer:

The graph of the function that increases faster for \(x > 1\) is \(f(x)\) (or \(\log_{2}x\)).