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

Evaluate

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

Evaluate

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

giving your answer in the form $\ln q$, 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 $1={\log }_{a}a$, show that

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

Write the following as a single logarithm

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

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

$2\ln 3^4+\ln 3^3-\ln 9$

The diagram below shows the length of three sides of a triangle, with each side measured in centimetres.

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

Show that

$2\ln x^3-3\ln x^2=0$

Solve the equation

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

Solve the equation

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

(i) On the same axes, sketch the graphs of $y=e^x$ and $y=e^{-x}$.

Label any points of intersection between each graph and the coordinate axes.

Write down the equations of any asymptotes.

(ii) Write down the equation of the line of reflection between the graphs of $y=e^x$ and   $y=e^{-x}$

Solve the equation

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

Evaluate

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

Evaluate

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

giving your answer in the form $\ln p$.

Solve the equation

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

Solve the equation

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

giving your answer to 3 significant figures.

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

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

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

Simplify

$2\ln 3^4+\ln 3^3-\ln 9$

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

Write

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

as a single logarithm.

(i) On the same axes, sketch the graphs of $y=e^x$ and $y=\ln x$

On each graph, label any points where the graph intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

(ii) Write down the line of reflection between the graphs $y=e^x$ and $y=\ln x$.

Solve the equation

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

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

Solve the equation

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

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

A ship sets sail from a harbour.

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

Find the direct distance between the ship and the harbour at this time giving your answer in the form $\left(p\ln 3\right)$km.

By writing 5 as $5\ln e$, show that 

$5\ln 2+5$

can be written as $5\ln 2e$

Solve the equation

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

giving your answers correct to 3 significant figures.

Solve the equation

$2{\log }_x\left(x+2\right)=3$

giving your answer correct to 3 significant figures.

Evaluate

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

Evaluate

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

giving your answer in the form $\ln q$.

Solve the equation

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

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

Solve the following equations, giving your answers correct to 3 significant figures.

$8e^{3x^2-1}=12$

$e^{3x}-42=2e^x\left(6e^x-7\right)$

Show that

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

Write the following as a single logarithm

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

On the same axes, sketch the graphs of $y=e^{2x}$ and $y=\frac{1}{2}\ln x$

On each graph, label any points where the graph intersects the coordinate axes.

Write down the equations of any asymptotes for each graph.

Explain the significance of the line $y=x$.

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

Α 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 $\left(3\ln 2\right)$ 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 $\alpha \ln 2$.

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

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

Without using a calculator, show that

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