Question

in the problem, $f(x)=\log_{2}x$ and $g(x)=\log_{10}x$.
for $x>1$, the graph of which function increases faster?
f
g
they increase at the same rate.

Explanation:

Step1: Convert logs to natural log

Use change of base formula: $\log_b x = \frac{\ln x}{\ln b}$
$f(x)=\frac{\ln x}{\ln 2}$, $g(x)=\frac{\ln x}{\ln 10}$

Step2: Find derivatives (rate of change)

Differentiate using $\frac{d}{dx}\ln x=\frac{1}{x}$
$f'(x)=\frac{1}{x\ln 2}$, $g'(x)=\frac{1}{x\ln 10}$

Step3: Compare derivatives for $x>1$

Since $\ln 2 < \ln 10$, $\frac{1}{\ln 2} > \frac{1}{\ln 10}$. For $x>1$, $\frac{1}{x\ln 2} > \frac{1}{x\ln 10}$, so $f'(x) > g'(x)$.

Answer:

f