An investment of $5000$ euros increases by $3.2\%$ each year. The value, $V$ euros, after $t$ years is modelled by $V=5000(1.032)^t$.

The graphs of $y=e^x$ and $y=4-x$ intersect at the point $A$.

A phone battery is being charged. The percentage charge, $B$, after $t$ hours is modelled by $B(t)=100-80e^{-kt}$, where $k$ is a positive constant. After $2$ hours, the battery charge is $55\%$.

Consider the equation $2^x+2^{-x}=\dfrac52$.

Solve $\log_x 16+\log_4 x=3$, where $x>0$ and $x\neq 1$.

The function $f$ is defined by $f(x)=ae^{kx}+b$, where $a>0$ and $k>0$. The graph of $f$ has horizontal asymptote $y=2$ and passes through the points $(0,5)$ and $(\ln 4,14)$.

Positive variables $x$ and $y$ are related by $y=Cx^n$, where $C$ and $n$ are constants. It is known that $y=6$ when $x=1$, and $y=48$ when $x=4$.

Let $f(x)=a^x$, where $0<a<1$. The point $(2,\frac{1}{9})$ lies on the graph of $f$.

A physical quantity $y$ is modelled by $y=Cx^n$, where $C$ and $n$ are positive constants. Two experimental readings are $y=10.4$ when $x=3$, and $y=83.1$ when $x=12$.

Consider the inequality $\ln(x+1)>\frac{1}{2}x-1$, where $x>-1$.

Let $p(x)=\log_2(x+3)+\log_2(7-x)$.

Let $f(x)=a^x$, where $a>1$. The point $(3,5)$ lies on the graph of $f$.

The function $f$ is defined by $f(x)=5-4e^{-x}$, for $x\in\mathbb{R}$.

The temperature $T$, in degrees Celsius, of a liquid after $t$ minutes is modelled by

where $C$ and $k$ are positive constants. Initially the temperature is $80^\circ\text{C}$. After $4$ minutes the temperature is $49^\circ\text{C}$.

Let $h(x)=\log_3(x-1)+\log_3(10-x)$.

Let $f(x)=\log_a x$, where $a>1$. The point $(8,3)$ lies on the graph of $f$. A second function is defined by $h(x)=\log_2(x-3)+1$.

Let $f(x)=\ln(e^x-1)$, for $x>0$.

A cup of soup is left in a room where the temperature is $18^\circ\text{C}$. The temperature of the soup, $T^\circ\text{C}$, $t$ minutes after it is left in the room is modelled by

where $A$ and $k$ are positive constants. Initially the temperature of the soup is $92^\circ\text{C}$, and after $5$ minutes its temperature is $61^\circ\text{C}$.

The function $f$ is defined by

where $a$ is a positive constant. The graph of $f$ passes through the point $(6,7)$.

The function $f$ is defined by

Let $P$ be the point of intersection of the graph of $f$ with the line $y=x$.

The mass, $M$ grams, of a sample of yeast is modelled by

where $t$ is measured in hours. The mass is $12.0$ grams initially and $19.5$ grams after $4$ hours.

A cup of liquid is removed from a heater and placed in a room. Its temperature, $T$ degrees Celsius, $t$ minutes later is modelled by

The graph shows the cooling curve and its horizontal asymptote.

The light level, $L$ lux, in a greenhouse after automatic shading is activated is modelled by

where $t$ is measured in minutes. The graph shows $L$ approaching a limiting light level.

Positive variables $x$ and $y$ are related by $y=Cx^n$, where $C$ and $n$ are constants. A graph of $Y=\ln y$ against $X=\ln x$ is a straight line passing through the points $(\ln 2,\ln 12)$ and $(\ln 8,\ln 96)$.

For $x\in\mathbb{R}$, define

Let $p>0$. Consider the equation

The function $f$ is defined by

For $a>0$, define