6.04 Solving exponential equations
Solve the following equations:
x^{5} = 3^{5}
8^{ - 7 } = x^{ - 7 }
x^{3} = \left(\dfrac{8}{5}\right)^{3}
x^{ - 7 } = \dfrac{1}{6^{7}}
3 \left(x^{ - 9 }\right) = \dfrac{3}{2^{9}}
x^{\frac{1}{3}} = \sqrt[3]{6}
\sqrt[3]{5} = x^{\frac{1}{3}}
\dfrac{1}{2^{5}} = x^{-5}
Solve the following exponential equations:
4^{x} = 4^{8}
3^{x} = 3^{\frac{2}{9}}
3^{x} = 27
7^{x} = 1
8^{x} = \dfrac{1}{8^{2}}
3^{y} = \dfrac{1}{27}
7 \left(4^{x}\right) = \dfrac{7}{4^{3}}
5^{x} = \sqrt[3]{5}
30^{n} = \sqrt[3]{30}
10^{x} = 0.01
Consider the following equations:
Rewrite each side of the equation with a base of 2.
Hence, solve for x.
8^{x} = 4
16^{x} = \dfrac{1}{2}
\dfrac{1}{1024} = 4^x
\left(\sqrt{2}\right)^{x} = \sqrt[5]{32}
Solve the following exponential equations:
\left(\sqrt{6}\right)^{y} = 36
\left(\sqrt{2}\right)^{k} = 0.5
9^{y} = 27
3^{ 5 x - 10} = 1
25^{x + 1} = 125^{ 3 x - 4}
\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}
\left(\dfrac{1}{9}\right)^{x + 5} = 81
\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}
\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}
8^{x + 5} = \dfrac{1}{32 \sqrt{2}}
30 \times 2^{x - 6} = 15
2^{x} \times 2^{x + 3} = 32
3^{x} \times 9^{x - k} = 27
a^{x + 1} = a^{3} \sqrt{a}
3^{x^{2} - 3 x} = 81
27 \left(2^{x}\right) = 6^{x}
Consider the following equations:
Determine the substitution, m that would reduce the equation to a quadratic.
Hence, solve the equation for x.
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.