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?

Explanation:

Step1: Use change of base formula

Rewrite both logs:
$f(x)=\frac{\ln x}{\ln 2}$, $g(x)=\frac{\ln x}{\ln 10}$

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 derivative magnitudes

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

Answer:

The graph of $f(x)=\log_2 x$ increases faster for $x>1$.