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 $y = Ce^{kx}$.

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 $y = ab^x$.

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 $y = ax^b$.

Find the linearized form of $y = A e^{k t}$ 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 $y = a b^x$.