Laws of Logarithms (AQA AS Maths: Pure): Exam Questions

Express

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

in the form

${\log }_{a}y$

where $a,b,c>0$ and $y$ is an expression in terms of $a$, $b$ and $c$ that you should find.

The diagram below shows the length of three sides of a triangle.

Find the perimeter of the triangle, giving your answer in the form

$2\ln b$

where $b$ is an integer to be found.

Simplify fully the expression

$2\ln \left(x^3\right)-3\ln \left(x^2\right)$

Solve the equation

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

giving your answer to 3 significant figures.

(i) On the same set of coordinate axes, sketch the curves $y={\mathrm{e}}^x$ and $y=\ln x$.

Label clearly the coordinates of any points where the curves meet the coordinate axes.

(ii) State the equation of any asymptotes and the equation of the line of reflection between the two curves.

A ship sets sail from a harbour.

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

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

Express

$5\ln 2+5$

in the form $5\ln k$ where $k$ is an irrational number to be found.

Solve

$8{\mathrm{e}}^{\left(3x^2-1\right)}=12$

giving your answer(s) to 3 significant figures.

Solve the equation

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

giving your answer(s) in the form $\ln a$ where $a$ is a constant to be found.

Express

$2\log x+3\log \left(x+1\right)-\log \left(4\left(x+2\right)\right)$

in the form

$\log \mathrm{f}\left(x\right)$

where $\mathrm{f}\left(x\right)$ is a function that you should state.

Solve the equation

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

giving your answer(s) in the form ${\log }_{5}k$ where $k$ is a constant to be found.

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 positive integers to be found.

Solve the equation

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

Solve the equation

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

giving your answer(s) in the form ${\log }_{5}a$, where $a$ is a constant to be found.

Show that

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

can be expressed in the form

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

where $x>1$.