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Wize High School Grade 12 Pre-Calculus Textbook > Rates of Change

Rates of Change of Logarithmic Functions

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Rates of Change of Logarithmic Functions

The rate of change of a logarithmic function can be found using algebraic and graphical methods.

Example

A certain species growth is modelled by the functionP(t)=15log⁡t+4P(t)=15\log{t}+4 , where P(t)P(t) is the population of the species at time tt, in years. How fast is the population growth changing when t=13t=13 years?

Algebraic Method


minst=(15log⁡(t+h)+4)−(15log⁡t+4)(t+h)−tLet h=0.001=15log⁡(13+0.001)−15log⁡(13)0.001=0.501 species/year\begin{array}{rcl} m_{inst}&=&\dfrac{\Big(15\log{(t+h)}+4\Big)-\Big(15\log{t}+4\Big)}{(t+h)-t}&&\colorTwo{\footnotesize{\text{Let}~h=0.001}}\\\\ &=&\dfrac{15\log(13+0.001)-15\log(13)}{0.001}\\\\ &=&0.501~\text{species}/\text{year} \end{array}


Graphical Method


In a graphing calculator:

Y=15log⁡x+4 + GRAPH + 2nd + PGRM + 5:Tangent + 13 + ENTER\boxed{Y=15\log{x}+4}~+~\boxed{\text{GRAPH}}~+~\boxed{2^{\text{nd}}}~+~\boxed{\text{PGRM}}~+~\boxed{5:\text{Tangent}}~+~\boxed{13}~+~\boxed{\text{ENTER}}


The tangent line is y=0.501x+14.19473y=0.501x+14.19473.

The IRC is equivalent to the slope.

Therefore, the IRC is 0.501 species/year0.501~\text{species}/\text{year}.
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Example: Rates of Change of Logarithmic Functions

A certain bacterium decays logarithmically. The amount left, in mg\text{mg}, can be determined by A(t)=−34log⁡2t+2A(t)=-34\log_{2}t+2, where tt is in weeks.

How fast is the bacterium decaying when t=4t=4?

minst=(−34log⁡2(t+h)+2)−(−34log⁡2t+2)(t+h)−tLet h=0.001=−34log⁡2(4+0.001)+34log⁡2(4)0.001≈−12.26 mg/week\begin{array}{rcl} m_{inst}&=&\dfrac{\Big(-34\log_{2}{(t+h)}+2\Big)-\Big(-34\log_2{t}+2\Big)}{(t+h)-t}&&\colorTwo{\footnotesize{\text{Let}~h=0.001}}\\\\ &=&\dfrac{-34\log_2(4+0.001)+34\log_2(4)}{0.001}\\\\ &\approx&-12.26~\text{mg}/\text{week} \end{array}

Practice: Rates of Change of Logarithmic Functions

Let f(x)=4log⁡3(x−3)+1f(x)=4\log_3(x-3)+1. Determine the instantaneous rate of change when x=6x=6.

Practice: Rates of Change of Logarithmic Functions

A radioactive substance decays logarithmically. It can be modelled by the function A(t)=−3log⁡52tA(t)=-3\log_5{2t}, where A(t)A(t) is the amount of the substance in mg\text{mg} after tt days.

Practice: Rates of Change of Logarithmic Functions

If the instantaneous rate of change for the function f(t)=2log⁡3t+1f(t)=2\log_3{t}+1 is 1010, what is tt? Let h=0.001h=0.001.



Extra Practice
Rates of Change of Logarithmic Functions
Practice: Rates of Change of Logarithmic Functions
If the instantaneous rate of change for the function f(t)=2log⁡3t+1f(t)=2\log_3{t}+1 is 1010, what is tt? Let h=0.001h=0.001.
Rates of Change of Logarithmic Functions
Practice: Rates of Change of Logarithmic Functions
A radioactive substance decays logarithmically. It can be modelled by the function A(t)=−3log⁡52tA(t)=-3\log_5{2t}, where A(t)A(t) is the amount of the substance in mg\text{mg} after tt days.
Rates of Change of Logarithmic Functions
Practice: Rates of Change of Logarithmic Functions
A radioactive substance decays logarithmically. It can be modelled by the function A(t)=−3log⁡52tA(t)=-3\log_5{2t}, where A(t)A(t) is the amount of the substance in mg\text{mg} after tt days.
Rates of Change of Logarithmic Functions
Practice: Rates of Change of Logarithmic Functions
Let f(x)=4log⁡3(x−3)+1f(x)=4\log_3(x-3)+1. Determine the instantaneous rate of change when x=6x=6.
Rates of Change of Logarithmic Functions
Practice: Rates of Change of Logarithmic Functions
Let f(x)=4log⁡3(x−3)+1f(x)=4\log_3(x-3)+1. Determine the instantaneous rate of change when x=6x=6.