Laws of Logarithms (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

Evaluate

${\text{log}}_{2}4+{\text{log}}_{3}27-{\text{log}}_{4}4$

Evaluate

$3\text{ln}2+\frac{1}{2}\text{ln}81-2\text{ln}3$

giving your answer in the form $\text{ln}q$, where $q$ is an integer to be found.

Solve the following equations, giving your answers in exact form.

$e^x=5$

$3e^{2x}=9$

$e^{2x-1}=4$

By writing $1={\text{log}}_{a}a$, show that

$1+2{\text{log}}_{a}b+3{\text{log}}_{a}c={\text{log}}_{a}a{b}^2{c}^3$

Write the following as a single logarithm

$2{\text{log}}_{a}6+3{\text{log}}_{a}2-{\text{log}}_{a}4$

Write the following in the form $a\text{ln}b$, where a and b are integers to be found.

$2\text{ln}{3}^4+\text{ln}{3}^3-\text{ln}9$

The diagram shows a triangle. The length of each of its three sides is measured in centimetres.

Work out the perimeter of the triangle, giving your answer in the form $2\text{ln}b$, where $b$ is an integer to be found.

Solve the equation

$5^{2x}-25=0$

Solve the equation

$3^{2x-1}=4^3+4^2+1$

Solve the equation

${\text{log}}_x\left(5x-6\right)=2$

Evaluate

${\text{log}}_2{8}^2+3{\text{log}}_{2}16-2{\text{log}}_2{2}^5$

Evaluate

$3\text{ln}2+2\text{ln}5-\frac{1}{2}\text{ln}10000$

giving your answer in the form $\text{ln}p$.

By writing 5 as $5\text{ln}e$, show that

$5\text{ln}2+5$

can be written as $5\text{ln}2e$.

Evaluate

$4{\text{log}}_{3}729+3{\text{log}}_2{64}^2-3\text{log}100+\text{ln}{e}^6$

Evaluate

$\frac{1}{2}\text{ln}196+\frac{1}{3}\text{ln}125+\frac{1}{4}\text{ln}81+\frac{1}{5}\text{ln}32$

giving your answer in the form $\text{ln}q$.

Solve the equation

$4^{3x+2}={16}^{x+6}$

Solve the equation

$4^{2x+3}-8=92$

giving your answer correct to 3 significant figures.

Solve the following equations, giving your answers in exact form.

$4e^{3x-2}=12$

$3e^{2x}+8=14e^x$

Simplify

$2\text{ln}{3}^4+\text{ln}{3}^3-\text{ln}9$

giving your answer in the form $a\text{ln}b$, where $a$ and $b$ are integers to be found.

Write

$2{\text{log}}_{a}x+3{\text{log}}_a\left(x+1\right)-{\text{log}}_{a}4\left(x+2\right)$

as a single logarithm.

Solve the equation

$5^{2x}-8\times 5^x+12=0$

giving your answers in the form ${\text{log}}_{a}b$.

Solve the equation

$6\times 3^{x-1}=6^{2x}$

giving your answer in the form $\frac{\text{ln}a}{\text{ln}b}$, where $a$ and $b$ are integers to be found.

A ship sets sail from a harbour.

After some time, the ship's position is $\left(4\text{ln}3\right)$ km east of the harbour and $\left(3\text{ln}3\right)$ km north of the harbour.

Find the direct distance between the ship and the harbour at this time, giving your answer in the form $\left(p\text{ln}3\right)$ km, where $p$ is an integer to be found.

Solve the equation

${\text{log}}_3\left(x+4\right)=4+2{\text{log}}_{3}x$

giving your answer correct to 3 significant figures.

Show that $4-\text{ln}16$ can be written in the form $4\text{ln}\left(\frac{e}{2}\right)$.

How many real solutions does the equation have? Justify your answer.

$3{\text{log}}_x\left(x+1\right)=\text{ln}{e}^3$

Solve the equation

$2\times 5^{2x+1}+21=41\times 5^x$

giving your answers in the form ${\text{log}}_{a}b$, where $a$ and $b$ are rational numbers to be found.

Show that

$2{\text{log}}_{3}x+{\text{log}}_3\left(x^2-1\right)-2{\text{log}}_3\left(x+1\right)\equiv {\text{log}}_3\frac{x^2\left(x-1\right)}{\left(x+1\right)}$

Write the following as a single logarithm

$2{\text{log}}_p\left(x+1\right)+3{\text{log}}_p\left(x-1\right)-{\text{log}}_p\left(x^2-1\right)$

A triangle is drawn inside a circle such that one side of the triangle is the diameter and all three vertices of the triangle lie on the circumference.

The radius of the circle is $\left(3\text{ln}2\right)$ cm.

The two smallest angles in the triangle are $\alpha$ and $\beta$ respectively, where $\beta =2\alpha$.

Find all three sides of the triangle, giving your answers in the form $a\text{ln}2$.

Without using a calculator, show that

${\text{log}}_{4}8={\text{log}}_{9}27$