find the minimum and maximum values of f(x)=log 2x and g(x)=log 10x on the given domain. 3000,7000 the…

Explanation:

Step1: Recall logarithmic function property

The functions $y = \log_2x$ and $y=\log_{10}x$ are increasing functions. So, for a closed - interval domain $[a,b]$, the minimum value occurs at $x = a$ and the maximum value occurs at $x = b$.

Step2: Calculate minimum of $f(x)=\log_2x$

For $f(x)=\log_2x$ and $x\in[3000,7000]$, when $x = 3000$, $f(3000)=\log_23000=\frac{\ln3000}{\ln2}\approx\frac{8.006367}{0.693147}\approx11.551$.

Step3: Calculate maximum of $f(x)=\log_2x$

When $x = 7000$, $f(7000)=\log_27000=\frac{\ln7000}{\ln2}\approx\frac{8.857332}{0.693147}\approx12.774$.

Step4: Calculate minimum of $g(x)=\log_{10}x$

For $g(x)=\log_{10}x$ and $x\in[3000,7000]$, when $x = 3000$, $g(3000)=\log_{10}3000=\log_{10}(3\times10^{3}) = 3+\log_{10}3\approx3 + 0.477=3.477$.

Step5: Calculate maximum of $g(x)=\log_{10}x$

When $x = 7000$, $g(7000)=\log_{10}7000=\log_{10}(7\times10^{3})=3+\log_{10}7\approx3 + 0.845 = 3.845$.

Answer:

The minimum and maximum values of $f(x)=\log_2x$ are approximately $11.551$ and $12.774$ respectively. The minimum and maximum values of $g(x)=\log_{10}x$ are approximately $3.477$ and $3.845$ respectively.