IB Mathematics Applications & Interpretation Question Type 2: Converting Exponential and Logarithmic Models Into Linearized Relations Exercises

If ${\log }_{10}y$ plotted against $x$ is a straight line with slope $0.4$ and intercept $0.7$, write the exponential model for $y$ in the form $y=a{b}^x$.

Express the model $y=4e^{rx}$ in a linear form relating $\ln y$ to $x$.

Given the linear relation $\ln y=2x+1$, express $y$ in terms of $x$ in the form $y=C{e}^{kx}$.

Given that plotting $\ln y$ vs $t$ for a decay process yields $\ln y=-0.02t+3.5$, determine the initial value and decay constant in $y=y_0{e}^{-kt}$.

Given the exponential model $y=5\times 3^x$, show that ${\log }_{10}y={\log }_{10}5+x{\log }_{10}3$.

The graph of $\ln y$ against $x$ passes through the points $(0,5)$ and $(2,20)$. Derive the exponential model $y=a{b}^x$.

A bacterial culture grows according to $N=200e^{0.3t}$, where $t$ is in hours. Write a linear equation for $\ln N$ versus $t$ and state its slope and intercept.

A researcher finds that plotting ${\log }_{10}y$ vs ${\log }_{10}x$ is linear with slope $2.5$ and intercept $-0.3$. Express the power-law model $y=a{x}^b$.

Find the linearized form of $y=A{e}^{kt}$ and identify its slope and intercept.

Convert the power law model $y=2x^{1.5}$ into a linear relation between $\ln y$ and $\ln x$.

Data plotted on a semi-log graph of $\ln y$ vs $x$ gives a straight line through points $(1,0.5)$ and $(4,2)$. Determine the exponential law $y=a{b}^x$.