Question Type 2: Converting exponential and logarithmic models into linearized relations Exercises

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

Given the exponential model $y = 5 imes 3^x$, show that $log_{10}y = log_{10}5 + x log_{10}3$.

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

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

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

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 $y = a b^x$.

Given the linear relation $ln y = 2x + 1$, convert back to the exponential form $y = Ce^{kx}$.

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 $y = a b^x$.

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 $y = y_0 e^{-kt}$.

Data plotted on a semi-log graph ($x$ linear, $y$ logarithmic) passes through $(0,5)$ and $(2,20)$. Derive the exponential model $y = ab^x$.

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 $y = ax^b$.