in the problems below, ( f(x)=log _{2} x ) and ( g(x)=log _{10} x ).\nhow are the graphs of ( f ) and ( g )…

in the problems below, ( f(x)=log _{2} x ) and ( g(x)=log _{10} x ).\nhow are the graphs of ( f ) and ( g ) similar? check all that apply.\nboth have a ( y )-intercept of 1.\nboth increase from left to right.\nboth have an asymptote of ( x = 0 ).\nboth have a domain of all real numbers.\nwhich point do the graphs of ( f ) and ( g ) have in common?\nfor ( x>1 ), the graph of which function increases faster?

in the problems below, ( f(x)=log _{2} x ) and ( g(x)=log _{10} x ).\nhow are the graphs of ( f ) and ( g ) similar? check all that apply.\nboth have a ( y )-intercept of 1.\nboth increase from left to right.\nboth have an asymptote of ( x = 0 ).\nboth have a domain of all real numbers.\nwhich point do the graphs of ( f ) and ( g ) have in common?\nfor ( x>1 ), the graph of which function increases faster?

Answer

Explanation:

Step1: Find the common point

For any logarithmic function (y = \log_{a}x), when (x = 1), (y=\log_{a}1 = 0). For (f(x)=\log_{2}x), when (x = 1), (f(1)=\log_{2}1=0). For (g(x)=\log_{10}x), when (x = 1), (g(1)=\log_{10}1 = 0).

Step2: Compare the growth rate for (x>1)

We know that the change - of - base formula for a logarithmic function (y=\log_{a}x=\frac{\ln x}{\ln a}). For (f(x)=\log_{2}x=\frac{\ln x}{\ln 2}) and (g(x)=\log_{10}x=\frac{\ln x}{\ln 10}), and (\ln 2\approx0.693), (\ln 10\approx2.303). The derivative of (y = \log_{a}x) using the formula (\frac{d}{dx}\log_{a}x=\frac{1}{x\ln a}). For (x > 1), the function (y=\log_{a}x) is increasing. Since (\ln 2<\ln 10), for (x>1), the function (f(x)=\log_{2}x) has a larger derivative.

Answer:

The common point is ((1,0)). For (x > 1), the graph of (f) increases faster.