Find the value of $b$.

Show that $S_{12}=12{\log }_{2}x-66$.

Find the average number of earthquakes in a year with a magnitude of at least $7.2$.

Find the value of a and of b.

Write an expression for $C$ in terms of $pH$.

Calculate the pH of the coffee.

Determine whether the unknown liquid is more or less acidic than the coffee. Justify your answer mathematically.

Find the value of $a$.

Find the value of $b$.

Given $0<M<8$, find the range for $N$.

Find the value of $a$.

Show that the sum of the infinite sequence is $4{\log }_{2}x$.

Find $r$.

Use the regression equation to estimate the value of y when x = 3.57.

The region R, is bounded by the graph of the function found in part (b), the x-axis, and the lines $x=1$ and $x=\alpha$ where $\alpha >1$. The area of R is $\sqrt{2}$.

Find the value of $\alpha$.

Solve .

Given that $S_{12}$ is equal to half the sum of the infinite geometric sequence, find $x$, giving your answer in the form $2^p$, where $p\in \mathbb{Q}$.

Show that $\text{lo}{\text{g}}_{r^2}x=\frac{1}{2}\text{lo}{\text{g}}_{r}x$ where $r,x\in {\mathbb{R}}^{+}$.

The relationship between x and y can be modelled using the formula y = kxⁿ, where k ≠ 0 , n ≠ 0 , n ≠ 1.

By expressing ln y in terms of ln x, find the value of n and of k.

Find the solution of ${\log }_{2}x-{\log }_{2}5=2+{\log }_{2}3$.

Express $y$ in terms of $x$. Give your answer in the form $y=p{x}^q$, where p , q are constants.

Find $d$, giving your answer as an integer.

An orchestra has a sound intensity of 6.4 × 10⁻³W m⁻² . Calculate the intensity level, $L$ of the orchestra.

A rock concert has an intensity level of 112 dB. Find the sound intensity, $S$.

Let $p=c^2$ and $q=c^3$. Find the value of $\sum _{n=1}^{20}{u}_n$.

Solve the equation $4^x+2^{x+2}=3$.

Find the $pH$ value for a solution in which the hydrogen ion concentration is $5.2\times {10}^{-8}$.

Find the hydrogen ion concentration in a solution with $pH 4.2$. Give your answer in the form $a\times 10k$ where $1\le a<10$ and $k$ is an integer.