1.8.7 Reduce y = ax^n and y = ab^x to linear form

Easy Medium

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Question 41

The intensity $I\,$ of radiation at a distance $d\,$ from a source is modeled by the equation

\frac{10^9}{d^3}, \quad d > 0

Sketch the graph of $log_{10} I$ against $log_{10} d$ . Show on your sketch the coordinates of the points of intersection of the graph with the axes.

[4]

ii.

A biologist monitors the growth of a bacterial colony. The relationship between the population size $P\,$ and time $t$ (in hours) is represented by a linear relationship between $log_5 P\,$ and $t$ . The graph of $log_5 P\,$ against $t\,$ is a straight line passing through the points $(0, 2)$ and $(- 10, 0)$ . Show that $P = ab^t$ where $a\,$ and $b\,$ are constants to be found.