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

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

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$x$.

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$x$.

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

On log–log axes (base 10), a straight line fit has gradient $-0.5$ and $y$-intercept $0.7$. Write the corresponding power law $y=a x^b$.

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 $x=5$.

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.ind $a$ and $b$).

Newton’s law of cooling states $T(t)-20 = (T_0-20)e^{-kt}$. A plot of $\ln[T(t)-20]$ versus $t$ is a straight line of gradient $-0.04$ and intercept $3$. Find $k$ and the initial temperature $T_0$.

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

x: 1, 2, 4

y: 3, 12, 48

Find $a$ and $b$.

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

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

In an experiment, pressure $P$ and volume $V$ of a gas are plotted on log–log axes (base 10), yielding a straight line of slope $-1.05$ and intercept $2.3$. Express $P$ as a function of $V$ and comment on the deviation from the ideal inverse law ($P\propto V^{-1}$).