IB Mathematics: Applications and Interpretation Quiz: Logarithmic Scales And Transformations

A set of bivariate data $(x, y)$ is believed to follow a power law relationship of the form $y=ax^b$. Which of the following transformations would result in a linear graph, and what would the gradient of that graph represent?

A population $P$ is modelled by the function $P(t) = A \cdot (10^{0.5})^t$, where $t$ is time in years. The data is linearized by plotting $\log_{10}(P)$ versus $t$. What is the gradient of the resulting line?

A student is modelling a relationship with the function $y = 4 \cdot (1.5)^x$. To linearize the data, they decide to plot $\ln(y)$ against $x$. What is the theoretical gradient of this linear graph, correct to 3 significant figures?

The growth of a bacterial colony is modelled by the exponential function $P = P_0 e^{kt}$, where $P$ is the population at time $t$ hours. To analyse the growth rate, a graph of $\ln(P)$ is plotted against $t$. The regression line has a gradient of 0.05. What is the correct interpretation of this gradient?

A researcher is studying the energy released by earthquakes, which can range from very small tremors to catastrophic events. Which of the following is the primary mathematical reason for using a logarithmic scale, like the Richter scale, to represent this data?

The relationship between two variables $x$ and $y$ is given by the power model $y=ax^k$. When $\log_{10}(y)$ is plotted against $\log_{10}(x)$, a straight line is formed which passes through the points $(0.5, 2.1)$ and $(1.5, 5.1)$. Find the value of the exponent $k$.

The relationship between the mass $M$ (in kg) and the wingspan $W$ (in m) of a species of bird is modelled by the equation $M = aW^b$. A researcher plots $\log_{10}(M)$ against $\log_{10}(W)$ and obtains a line of best fit with the equation $Y = 2.5X - 0.7$, where $Y = \log_{10}(M)$ and $X = \log_{10}(W)$. Find the value of $a$, correct to two decimal places.

A chemist has a solution with a pH of 5. She adds a substance that causes the hydrogen ion concentration, $[H^+]$, to decrease by a factor of 1000. What is the new pH of the solution?

An earthquake in region A was measured to have a magnitude of 6.8 on the Richter scale. A seismic event in region B was found to be 40 times less intense than the earthquake in region A. What is the magnitude of the seismic event in region B on the Richter scale, correct to one decimal place?

A biologist models the growth of a yeast culture using the equation $P(t) = A \cdot B^t$, where $P$ is the population at time $t$ hours. To linearize the data, the biologist plots $\ln(P)$ against $t$. The resulting line of best fit is given by the equation $Y = 0.15t + 4.20$, where $Y = \ln(P)$. What is the value of $A$ in the biologist's model, correct to the nearest integer?

An earthquake in region A was measured to have a magnitude of 6.8 on the Richter scale. A seismic event in region B was found to be 40 times less intense than the earthquake in region A. What is the magnitude of the seismic event in region B on the Richter scale, correct to one decimal place?

A dataset is modelled by the exponential function $f(x) = ab^x$. The data is linearized by plotting $\ln(y)$ against $x$. The resulting line passes through the points $(2, 5.5)$ and $(6, 8.3)$. Find the value of $b$ in the model, correct to two decimal places.

A scientist proposes a power law model $S = aT^b$ to relate the number of species $S$ on an island to its area $T$ in square kilometers. By plotting $\ln(S)$ versus $\ln(T)$, she obtains the regression line $Y = 0.25X + 1.8$, where $Y=\ln(S)$ and $X=\ln(T)$. Predict the number of species on an island with an area of 200 square kilometers, rounding your answer to the nearest whole number.

A power law model $y = ax^b$ is linearized by plotting $Y$ against $X$, where $Y=\ln(y)$ and $X=\ln(x)$. The equation of the resulting regression line is $Y = mX + c$. Which expression correctly gives the value of the coefficient $a$ from the original model?

The growth of a bacterial colony is modelled by the exponential function $P = P_0 e^{kt}$, where $P$ is the population at time $t$ hours. To analyse the growth rate, a graph of $\ln(P)$ is plotted against $t$. The regression line has a gradient of 0.05. What is the correct interpretation of this gradient?

A scientist collects data on a variable $y$ that depends on $x$. To determine the nature of the relationship, two transformations are performed. A plot of $\ln(y)$ versus $x$ has a Pearson's correlation coefficient of $r = 0.88$. A plot of $\ln(y)$ versus $\ln(x)$ has a Pearson's correlation coefficient of $r = -0.99$. Which of the following is the most appropriate conclusion?

A student is modelling a relationship with the function $y = 4 \cdot (1.5)^x$. To linearize the data, they decide to plot $\ln(y)$ against $x$. What is the theoretical gradient of this linear graph, correct to 3 significant figures?

It is observed for a certain physical phenomenon that when the logarithm of a quantity $P$ is plotted against the logarithm of a quantity $Q$, the result is a straight line with a negative slope. Which statement best describes the relationship between $P$ and $Q$?

A biologist models the growth of a yeast culture using the equation $P(t) = A \cdot B^t$, where $P$ is the population at time $t$ hours. To linearize the data, the biologist plots $\ln(P)$ against $t$. The resulting line of best fit is given by the equation $Y = 0.15t + 4.20$, where $Y = \ln(P)$. What is the value of $A$ in the biologist's model, correct to the nearest integer?

The sound level of a machine in a factory is 90 dB. A second, identical machine is turned on next to the first one, which doubles the total sound intensity. What is the new sound level in decibels, correct to one decimal place?