Show that $p=\frac{2}{3}$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

Find the value of $k$ and the value of $p$.

By solving the logistic differential equation, show that its solution can be expressed in the form

$kt=\ln \frac{P}{P_0}\left(\frac{N-P_0}{N-P}\right)$.

Sketch the graph of $f$ for $-30\le x\le 30$, clearly indicating the points of intersection with each axis and any asymptotes.

Hence or otherwise solve $\text{lo}{\text{g}}_3\left(\text{2}\text{sin}x\right)=\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)$ for .

Consider the equation ${\left(1251\right)}_n+{\left(30\right)}_n={\left(504\right)}_n+{\left(504\right)}_n$.

Find the value of $n$.

Show that $\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)=\text{lo}{\text{g}}_3\sqrt{\text{cos}2x+2}$.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Write down $d$ in the form $k \ln x$, where $k\in \mathrm{\mathbb{Q}}$.

Hence or otherwise, show that the series is convergent.

Given that $p>0$ and $S_{\infty }=3+\sqrt{3}$, find the value of $x$.

Show that $p=\frac{2}{3}$.

The sum of the first $n$ terms of the series is $\ln \left(\frac{1}{x^3}\right)$.

Find the value of $n$.

Solve .

Solve the equation ${\log }_2(x+3)+{\log }_2(x-3)=4$.

Write down the remainder when ${14}^{2022}$ is divided by $7$.

Convert ${\left(7A2\right)}_{16}$ to base $5$, where ${\left(A\right)}_{16}={\left(10\right)}_{10}$.

Find the remainder when ${15}^{1207}$ is divided by $13$.

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

Solve the simultaneous equations

$\text{lo}{\text{g}}_{2}6x=1+2\text{lo}{\text{g}}_{2}y$

$1+\text{lo}{\text{g}}_{6}x=\text{lo}{\text{g}}_6\left(15y-25\right)$.

The base $13$ number $1y93y25$ is divisible by $6$. Find the possible values of the digit $y$.

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

Use Fermat’s little theorem to find the remainder when ${14}^{2022}$ is divided by $17$.

Given that ${\log }_{10}\left(\frac{1}{2\sqrt{2}}\left(p+2q\right)\right)=\frac{1}{2}\left({\log }_{10}p+{\log }_{10}q\right),p>0,q>0$, find $p$ in terms of $q$.

Prove that a number in base $13$ is divisible by $6$ if, and only if, the sum of its digits is divisible by $6$.

Find the value of a and of b.

Solve the equation $2 \ln x= \ln 9+4$. Give your answer in the form $x=p{\mathrm{e}}^q$ where $p, q\in {\mathrm{\mathbb{Z}}}^{+}$.

Find the value of ${\int }_1^9\left(\frac{3\sqrt{x}-5}{\sqrt{x}}\right)dx$.

$y$-axis.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

The expression $\frac{3\sqrt{x}-5}{\sqrt{x}}$ can be written as $3-5x^p$. Write down the value of $p$.

Consider the curve with equation $y=(2x-1){\text{e}}^{kx}$, where $x\in \mathrm{ℝ}$ and $k\in \mathrm{\mathbb{Q}}$.

The tangent to the curve at the point where $x=1$ is parallel to the line $y=5{\text{e}}^{k}x$.

Find the value of $k$.

Show that $p=\pm \frac{1}{\sqrt{3}}$.

Write down an expression for $f^{-1}\left(x\right)$.

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

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

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

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.

State Fermat’s little theorem.

The area enclosed by the graph of $y=f\left(x\right)$ and the line $y=4$ can be expressed as $\text{ln}v$. Find the value of $v$.

Express $\frac{1}{f\left(x\right)}$ in partial fractions.

Hence find the exact value of $\underset{0}{\overset{3}{\int }}\frac{1}{f\left(x\right)}dx$, expressing your answer as a single logarithm.

Solve the equation ${\log }_3 \sqrt{x}=\frac{1}{2 {\log }_2 3}+{\log }_3\left(4x^3\right)$, where $x>0$.

Show that $a=8$.

Find the value of $p$ and the value of $q$.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

$x$-axis.

Write down the equation of the vertical asymptote of the graph of $f$.