Solve the inequality $3\times 2^{3x+2}<8$, giving your answer in terms of logarithms.

Use logarithms to solve the equation
$5^{x+3}=7^{x-1},$

giving the answer correct to $3$ significant figures.

The variables $x$ and $y$ satisfy the equation $y=K{x}^m$, where $K$ and $m$ are constants. The graph of $\ln y$ against $\ln x$ is a straight line passing through the points $(0, 2.0)$ and $(6, 10.2)$, as shown in the diagram. Find the values of $K$ and $m$, correct to $2$ decimal places.

Given that $y=2^x$, show that the equation $2^x+3\left(2^{-x}\right)=4$ can be written in the form $y^2-4y+3=0.$

Hence Solve the equation $2^x+3\left(2^{-x}\right)=4,$ giving the values of $x$ correct to $3$ significant figures where appropriate.

The polynomial $f\left(x\right)$ is defined by $f\left(x\right)=12x^3+25x^2-4x-12$.

Show that $f(-2)=0$ and factorise $f\left(x\right)$ completely.