4.07 The natural logarithm
Consider the function f \left( x \right) = e^{x}.
Complete the following table of values, correct to two decimal places:
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| f(x) |
Sketch the graph of f \left( x \right) = e^{x}.
Sketch the curves y = e^{x}, y = e^{x} + 3, and y = e^{x} - 4 on the same coordinate plane.
Use a calculator or other technology to approximate the each of the following values correct to four decimal places:
e^{4}
e^{ - 1 }
e^{\frac{1}{5}}
5 \sqrt{e}
\dfrac{4}{e}
\dfrac{8}{9 e^{4}}
Find the value of each of the following correct to four decimal places:
\ln 94
\ln 0.042
\ln 78^{4}
\ln \left( 18 \times 35\right)
Consider x=\ln 31. Find the value of x, correct to two decimal places.
Find the value of 5 \log_{e} e.
Is the value of \log_{e} 2 greater than or less than 1?
Evaluate each of the following expressions:
\ln e^{3.5}
\ln e^{4}
\sqrt{6} \ln \left(e^{\sqrt{6}}\right)
\ln \left(\dfrac{1}{e^{2}}\right)
Write the following equations in logarithmic form.
The population P of a city increases according to the formula P = 3000 e^{ k t} where t is measured in years and k is a constant.
Find the initial population.
Given the population increases to 8000 in 2 years find the exact value of k.
How many complete years will it take for the population to at least double?
The voltage V in volts across an electrical component decays according to the equation V = A e^{ - \frac{1}{2} t } where A is the initial voltage and t is the time in years. Find the value of t such that V is half of the initial voltage, correct to two decimal places.
The spread of a virus through a city is modelled by the function: N = \dfrac{15\,000}{1 + 100 e^{ - 0.5 t }}, where N is the number of people infected by the virus after t days.
How many people will have been infected after 3 days? Round your answer to the nearest whole number.
How many whole days will it take for at least 4000 people to be infected with the virus?