Rose Harrison, Clara Huizink, Aidan Sproat Clements and, Marlene Torres Skoumal Solutions for Exercise 17: Practice 6

A state surveys school traffic. At the end of $2010$, there were $25000$ cars per day taking children to school. After $t$ years the number of cars, $C$, was modelled by the exponential equation $C=25000\times {\mathrm{e}}^{0.07t}$.

Show that by the end of $2015$, there were $35477$ cars per day taking children to school.   [Use $e^{0.35}=1.419068$]

A state surveys school traffic. At the end of $2010$, there were $25000$ cars per day taking children to school. After $t$ years the number of cars, $C$, was modelled by the exponential equation $C=25000\times {\mathrm{e}}^{0.07t}$.

By taking logs of both sides, calculate the time until the number of cars taking children to school reaches $50000$.      [Use $\ln 2=0.6931$]

The time taken for an online video to go viral can be modelled by an exponential function, $H=15{\mathrm{e}}^{0.5t}$, where $H$ is the number of times the video is watched and $t$ is the number of hours since the first people shared the video. Describe what the value$15$ represents in this model.

The time taken for an online video to go viral can be modelled by an exponential function, $H=15{\mathrm{e}}^{0.5t}$, where $H$ is the number of times the video is watched and $t$ is the number of hours since the first people shared the video. By taking logs of both sides, calculate the number of hours before the video is watched $500000$ times.    [Use $\ln 10=2.302585, \ln 3=1.098612$]

The time taken for an online video to go viral can be modelled by an exponential function, $H=15{\mathrm{e}}^{0.5t}$, where $H$ is the number of times the video is watched and $t$ is the number of hours since the first people shared the video. By rewriting the exponential statement as a logarithmic statement, calculate the number of hours before the video is watched $1000000$ times.    [Use $\ln 10=2.302585093, \ln 3=1.098612, \ln 5=1.609438$]

Solve:

$2^{x+1}=4^{2x}$

Solve:

$3^{x+2}=9^{2x-2}$

If ${\log }_{a}x=p$, show that rewriting it as a logarithmic statement and solving the resulting equation proves the power rule for logarithms.