Exponential and logarithmic functions：練習問題

Evaluate \( \log_2 (64) + \log_{10} (0.001) \).

Find the derivative of $$f(x) = \frac{\ln x}{x}$$.

The variables \(x\) and \(y\) satisfy the relation \(y = \frac{k}{(x^2+1)^n}\), where \(k\) and \(n\) are positive constants. It is given that the graph of \(\ln y\) against \(\ln(x^2+1)\) is a straight line passing through the points \( (1, 5) \) and \( (4, -1) \). Find the value of \(n\).

Simplify the expression \( \frac{\log_4 16}{\log_2 8} + \log_5 (\frac{1}{25}) \).

A population of bacteria \(N\) grows according to the model \(N = N_0 e^{kt}\), where \(t\) is the time in hours and \(N_0, k\) are constants. If the population triples every 5 hours, find the value of \(k\) correct to 3 decimal places.

If \( \ln(x^2) = 6 \) and \( x > 0 \), find the exact value of \( x \) in terms of the constant \( e \).

Using the first three terms of the exponential series \( e^x \approx 1 + x + \frac{x^2}{2!} \), find an approximate value of \( e^{0.1} + e^{-0.1} \).

The variables \( x \) and \( y \) are related by the equation \( y = Ax^n \). When \( \ln y \) is plotted against \( \ln x \), a straight line is obtained which passes through the points \( (2, 5) \) and \( (4, 1) \). Determine the value of \( n \) and express the constant \( A \) in terms of \( e \).

A certain radioactive substance decays such that its mass \(m\) (in grams) at time \(t\) (in years) is given by the model \(m = 100(0.95^t)\).
(a) Find the initial mass of the substance when \(t = 0\).
(b) Find the time required for the mass to decay to half of its initial value. Give your answer correct to 2 decimal places.

The variables \(x\) and \(y\) satisfy the relation \(y = kx^n\), where \(k\) and \(n\) are positive constants. The graph of \(\ln y\) against \(\ln x\) is a straight line as shown in the figure. The line passes through the points \(A(2, 5)\) and \(B(4, 9)\).
(a) Express \(\ln y\) as a linear function of \(\ln x\).
(b) Find the values of \(n\) and \(\ln k\). Hence, find the value of \(k\) in terms of \(e\).