20.07 Further exponential equations (Extended)
Find the interval in which the solution of the following equations will lie:
Consider the following equations:
Rearrange the equation into the form x = \dfrac{\log A}{\log B}.
Evaluate x to three decimal places.
13^{x} = 5
5^{x} = \dfrac{1}{11}
3^{x} = 2
4^{x} = 6.4
\left(0.4\right)^{x} = 5
5^{x} + 4 = 3129
2^{ - x } = 6
27^{x} + 4 = 19\,211
Consider the equation 4^{ 2 x - 8} = 70.
Make x the subject of the equation.
Evaluate x to three decimal places.
For each of the following incorrect sets of working:
Which step was incorrect? Explain your answer.
Rearrange the original equation into the form a = \dfrac{\log A}{\log B}.
Evaluate a to three decimal places.
\begin{aligned} 9 ^ {a} &= 40 \\ \log9^{a} &= \log40 & (1)\\ a + \log 9 &= \log40 & (2)\\ a &= \log40 - \log9 & (3)\\ &\approx 0.648 & (4) \end{aligned}
\begin{aligned} 2 ^ {a} &= 89 \\ \log2^{a} &= \log89 & (1) \\ a \log 2 &= \log89 & (2)\\ a &= \log_{89} 2 & (3)\\ &\approx 0.154 & (4) \end{aligned}
A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by:
N(t)=2^tFind the time when the number of cells will reach the following amounts. Round your answers to two decimal places where necessary.
1024
3000
A population of mice, t months after initial observation, is modelled by:
P(t)=500(1.2^t)State the initial population.
By what percentage is the poulation increasing by each month?
Find the time when the population reaches 1500 to two decimal places.
A population of wallabies, t years after initial observation, is modelled by:
P(t)=800(0.85^t)State the initial population.
By what percentage is the poulation decreasing by each year?
Find the time when the population reaches 200 to two decimal places.
A microbe culture initially has a population of 900\,000 and the population increases by 40\% every hour. Let t be the number of hours passed.
Find the time when the population reaches 7\,200\,000 to three decimal places.