Question Type 4: Using log-log and semi-log variables for estimation of parameters Exercises

In an experiment, the pressure $P$ and volume $V$ of a gas are plotted on log–log axes (base 10), yielding a straight line with a slope of $-1.05$ and an intercept of $2.3$.

Express $P$ as a function of $V$ and comment on the deviation from the ideal inverse law ($P \propto V^{-1}$).

A graph of $\ln y$ versus $\ln x$ is a straight line with gradient $2$ and intercept $0.3$.

Find the value of $y$ when $x=5$.

The data below follow a power law $y=ax^b$:

Find $a$ and $b$.

In a model $y = a x^b$, it is observed that tripling $x$ causes $y$ to increase ninefold. Determine $b$.

A semi-log graph plots $y$ (vertical axis) against $\log_{10}x$ (horizontal axis) and gives a straight line of gradient $1.5$ with $y$-intercept $0.2$. Express $y$ as a function of $x$.

A radioactive sample decays according to $\ln N = -0.231t + 6$, where $t$ is in days. Determine its half-life.

A semi-log plot of $\ln y$ versus $x$ is a straight line with gradient $-0.3$ and $y$-intercept $1$. Find the formula for $y$ in terms of $x$.

On a log–log graph of $y$ versus $x$, the fitted line has gradient $2$ and vertical intercept $0.5$ (using common logarithms). Express $y$ as a function of $x$.

On a log–log graph (base 10), two data points lie at $(x, y) = (10, 1000)$ and $(100, 8000)$. Determine the parameters of the power model $y = a x^b$ (i.e. find $a$ and $b$).

On log–log axes (base 10), a straight line of best fit has a gradient of $-0.5$ and a vertical intercept of $0.7$.

Determine the corresponding power law relation in the form $y=ax^b$.

A semi-log plot (common log of $y$ versus $x$) has equation $\log_{10} y = 0.1x + 2$. Determine the doubling time of $y$.

Newton’s law of cooling states $T(t)-20 = (T_0-20)e^{-kt}$, where $T(t)$ is the temperature in degrees Celsius ($^\circ\text{C}$) at time $t$ minutes, $T_0$ is the initial temperature, and $k$ is a constant.

A plot of $\ln(T(t)-20)$ against $t$ is a straight line with gradient $-0.04$ and vertical intercept $3$.

Find the value of $k$ and the value of $T_0$.