Exponentials and logarithms：練習題

Evaluate \(\log_2\left(\frac{1}{16}\right)\).

A radioactive substance decays according to the model \(N = N_0 e^{-0.05t}\), where \(t\) is time in years. Find the time taken, to the nearest year, for the amount of substance to decrease to half of its initial value.

A student uses a logarithmic graph to model the relationship \( y = ax^n \). They plot \( \log_{10} y \) against \( \log_{10} x \) and obtain a straight line passing through the points \( (0, 2) \) and \( (2, 8) \). Find the values of \( a \) and \( n \).

Solve the equation \(\log_{10}(x+2) = 2\) for \(x\).

The relationship between two variables \(x\) and \(y\) is thought to be of the form \(y = kx^n\). When \(\log_{10} y\) is plotted against \(\log_{10} x\), a straight line is obtained with a gradient of \(-2\) passing through the point \((0, 3)\). Find the values of \(k\) and \(n\).

Using the laws of logarithms, solve the following equation for \(x\):
\(2\log_{10}(x) - \log_{10}(x-2) = 1\)

A curve is defined by the equation \(y = 2^{3x-1}\).
Find the exact coordinates of the point where this curve intersects the line \(y = 5\), giving your answer for the \(x\)-coordinate in terms of \(\ln 2\) and \(\ln 5\).

A researcher is studying the growth of a particular strain of bacteria in a controlled environment. The population of the bacteria, \( P \), at time \( t \) hours after the start of the experiment is modelled by the equation:
\( P = Ae^{kt} \)
where \( A \) and \( k \) are constants.

(a) Show that the relationship between \( \ln P \) and \( t \) can be written in the form \( \ln P = kt + \ln A \). [2 points]

(b) The researcher records the following data:
At \( t = 2 \), \( P = 148 \)
At \( t = 5 \), \( P = 403 \)
By substituting these values into the linear equation from part (a), or otherwise, calculate the values of \( A \) and \( k \). Give your answers to 3 significant figures. [4 points]

(c) According to the model, find the time, in hours, for the initial population to triple. Give your answer to 2 decimal places. [2 points]