Exponential & Logarithmic functions & their graphs: notes and practice questions

A freshly baked cake is taken out of an oven and placed on a wire rack to cool in a kitchen.

The temperature of the cake, $T^\circ\text{C}$, after $t$ minutes is modelled by the function $T(t) = 21 + 144\text{e}^{-0.035t}$, for $t \ge 0$.

(a) Find the initial temperature of the cake.

(b) Find the temperature of the cake after 20 minutes.

(c) Write down the temperature of the kitchen.

A biologist is modelling the growth of two different bacterial colonies. The first colony, A, grows such that its population at time $x$ is given by $P_A(x) = a^x$, where $a$ is a growth factor and $x \ge 0$. The second colony, B, grows linearly such that its population at time $x$ is $P_B(x) = x$.

Consider the cases where the growth factor $a = 2$ and $a = 10$. On the same set of axes, sketch the following three graphs for $x \ge 0$:

$y = 2^x$

$y = 10^x$

$y = x$

Clearly label each graph with its equation and state the coordinates of any non-zero $y$-axis intercepts.

In parts (b) and (c), consider the case where the growth factor $a = e$.

Use calculus to find the minimum value of the expression $e^x - x$, justifying that this value is a minimum.

Hence deduce that $e^x > x$ for all $x \in \mathbb{R}$.

There exist values of $a$ for which the graph of $y = a^x$ and the line $y = x$ have different numbers of intersection points. The following table gives three intervals for the value of $a$.

By investigating the graph of $y = a^x$ for different values of $a$, write down the values of $p, q$ and $r$.

In parts (e) and (f), consider $a \in \mathbb{R}^+, a \neq 1$.

For $1.4 \leq a \leq 1.5$, a value of $a$ exists such that the line $y = x$ is a tangent to the graph of $y = a^x$ at a point P.

Find the exact coordinates of P and the exact value of $a$.

Write down the exact set of values for $a$ such that the graphs of $y = a^x$ and $y = x$ have

(i) two intersection points;

(ii) no intersection points.

Consider the functions $f(x) = e^{2x}$ and $g(x) = x+c$, where $c$ is a real constant.

(a) Write down an expression for $(f \circ g)(x)$.

(b) Given that $(f \circ g)(\ln 3) = 36$, find the value of $c$.

Consider the functions $f(x) = 5 - x^2$, $g(x) = \frac{3}{x-2}$, and $h(x) = e^x + 1$.

(a) Find the range of $f(x)$.

(b) Find the range of $g(x)$.

(c) Find the range of $h(x)$.

(d) Find an expression for $(g \circ f)(x)$.

(e) Solve the equation $(g \circ f)(x) = -1$.

(f) Solve the inequality $(h \circ f)(x) < e+1$.

The functions $f$ and $g$ are defined by

$f(x) = 2\ln x$, where $x > 0$

$g(x) = \ln(k(x-2) )$, where $x > 2, k \in \mathbb{R}^+$.

The graphs of $y = f(x)$ and $y = g(x)$ intersect at two distinct points.

(a) State the equation of the vertical asymptote to the graph of $y = g(x)$.

(b) (i) Show that, at the points of intersection, $x^2 - kx + 2k = 0$.

(b) (ii) Hence show that $k^2 - 8k > 0$.

(b) (iii) Find the range of possible values of $k$.

The graphs intersect at $x=p$ and $x=q$, where $p<q$.

(c) In the case where $k=10$, find the value of $q-p$. Express your answer in the form $a\sqrt{b}$, where $a, b \in \mathbb{Z}^+$.

A chemical reaction is monitored over time. The concentration of a reactant, $C$, in $\text{mol dm}^{-3}$, at time $t$ minutes, is modelled by the function $C(t) = k + Ae^{bt}$.

Initially, at $t=0$, the concentration of the reactant was $15.0\,\text{mol dm}^{-3}$. After $10$ minutes, the concentration had dropped to $7.0\,\text{mol dm}^{-3}$. It is known that the concentration approaches a minimum value of $2.0\,\text{mol dm}^{-3}$ as time increases.

(a) Find the value of

(i) $A$

(ii) $b$

(b) Use the model to estimate the concentration of the reactant after it has been reacting for $25$ minutes.

(c) Write down the equation of the horizontal asymptote of the graph of $C$.

(d) State the meaning of the asymptote found in part (c) in the context of this problem.

(e) Find an expression for the rate of change of the concentration of the reactant after $t$ minutes.

(f) Find an expression for $\frac{d^2C}{dt^2}$.

(g) Hence explain how the concentration of the reactant will vary if the reaction is left for a long time.

Consider the functions $f(x) = k - (x-h)^2$ and $g(x) = \ln(x-1) + 2$ where $h, k \in \mathbb{R}$.

The graphs of $f$ and $g$ have a common tangent at $x=2$.

(a) Find $g'(x)$.

(b) Show that $h = \frac{5}{2}$.

(c) Hence, find the value of $k$.

The following table shows the annual revenue of a tech startup, Quantum Innovations, $t$ years after its launch in 2015.

A data analyst uses linear regression to model the revenue of Quantum Innovations using these data.

The analyst's model is $R = at + b$.

(a)(i) Write down the value of $a$ and the value of $b$.

(a)(ii) Interpret, in context, the value of $a$.

(b) The analyst uses this model to predict the revenue of Quantum Innovations in the year 2030, where $t = 15$, and calculates a revenue of approximately $11.8$ million USD.

Comment on the reliability of the analyst's prediction.

(c)(i) A financial expert, Elena, develops an exponential model for Quantum Innovations' future revenue.

In this model, $R_E(t) = 1.6(1.08)^t$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Elena's model to predict the revenue of Quantum Innovations in the year 2035.

(c)(ii) Interpret, in context, the value $1.08$ in Elena's model.

(d) Another financial expert, Carlos, develops a third model for Quantum Innovations' revenue.

In this model, $R_L(t) = \frac{20}{1+21e^{-0.15t}}$ represents the revenue in millions of USD $t$ years after 2015, where $10 \le t \le 25$.

Use Carlos's model to predict the revenue of Quantum Innovations in the year 2035.

(e) Determine the year in which the difference between the predictions from Elena's model and Carlos's model is greatest.

(f)(i) Find the value of

$R_E'(18)$;

(f)(ii) Find the value of

$R_L'(18)$.

(g) Compare and interpret, in context, the values of $R_E'(18)$ and $R_L'(18)$.

Consider the function f defined by $f(x) = \ln(x^2 - 3)$ for $x > \sqrt{3}$.

The following diagram shows part of the graph of f which crosses the x-axis at point A, with coordinates $(p, 0)$. The line L is the tangent to the graph of f at the point B.

(a) Find the exact value of $p$.

(b) Given that the gradient of L is $1$, find the x-coordinate of B.

A function $f$ is defined by $f(x) = \frac{\ln(x)}{x}$, for $x > 0$.

The following diagram shows part of the graph of $f$.

(a) Find the coordinates of the x-intercept of the graph of $f$.

(b) Find $f'(x)$.

The graph of $f$ has a local maximum at point M.

(c) Hence, find the exact coordinates of M.

(d) (i) Show that $f''(x) = \frac{2\ln(x) - 3}{x^3}$.

The graph of $f$ has a point of inflection at point P.

(d) (ii) Hence, find the exact coordinates of P.

A pharmaceutical company is testing a new drug. The concentration of the drug in a patient's bloodstream, $C$, in mg/L, $t$ hours after administration, is modeled by the function $C(t) = 10t \text{e}^{-0.5t}$, for $0 \le t \le 10$.

Sketch the graph of $C(t)$ on the grid below.

(b) Find the time, in hours, at which the concentration of the drug in the bloodstream is at its maximum.

The amount of a certain pollutant, in tonnes, in a lake $t$ weeks after a major clean-up operation began, can be modelled by $P(t) = 5 + 75e^{-0.1t}$, $t \ge 0$.

Find the initial amount of pollutant in the lake.

Find the amount of pollutant in the lake five weeks after the clean-up operation began.

Write down the value of $P'(5)$.

Interpret the meaning of your answer to part (b) in the given context.

The clean-up operation is considered successful when the amount of pollutant in the lake is below 20 tonnes.

Find the least possible integer value of $k$, in weeks, for which the amount of pollutant is below 20 tonnes.

As the clean-up operation continues indefinitely, the amount of pollutant in the lake approaches a constant level.

Find this constant level of pollutant.

Find the limit of $P'(t)$ as $t$ approaches infinity.

A chemical spill introduces a pollutant into a lake. The concentration of the pollutant, $C$, in micrograms per litre ($\mu g/L$), $t$ days after the spill, is modelled by the function $C(t) = C_0 e^{-kt}$.

Initially, the concentration of the pollutant in the lake is $800 \, \mu g/L$.

(a) Write down the initial concentration of the pollutant.

After 5 days, the concentration is measured to be $350 \, \mu g/L$.

(b) Calculate the value of $k$.

(c) Determine the number of days it takes for the concentration to fall to $100 \, \mu g/L$.

A scientist is studying the growth of a certain bacterial culture. The population, in thousands, at time $t$ hours is modeled by the function $P(t) = e^t - 4t - 5$.

On the following axes, sketch the graph of $P(t)$ for $-2 \leq t \leq 4$.

Another bacterial culture, observed under different conditions, has its population modeled by the function $Q(t) = e^{2t} - 8t - 12$.

The graph of $Q(t)$ is obtained from the graph of $P(t)$ by a horizontal stretch with scale factor $k$, followed by a vertical translation of $c$ units.

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

(a) A new species of invasive algae is introduced into a lake. Its biomass, $B(t)$ in kg, at time $t$ weeks, is modelled by the function $B(t) = e^{0.5t} - 5$. Simultaneously, the concentration of a specific nutrient, $N(t)$ in arbitrary units, crucial for the algae's growth, is modelled by $N(t) = 2 \ln(t+1)$. Scientists are interested in finding the time(s) when the algae's biomass equals the nutrient concentration.

(i) On the same set of axes, sketch the graphs of $y = e^{0.5x} - 5$ and $y = 2 \ln(x+1)$ for $x \ge 0$. Clearly label any axis intercepts and the intersection point(s).

(ii) Hence, use your GDC to find the value of $t$ when the algae's biomass equals the nutrient concentration. Give your answer to three significant figures.

(a) A scientist is studying the population sizes of two bacterial cultures, A and B, in a controlled environment. The population sizes, in thousands, are modelled by the functions $f(x) = \ln(x+1)$ and $g(x) = \ln(5-x)$ respectively, where $x$ is the time in hours. Sketch the graphs of $f(x)$ and $g(x)$ on the same set of axes for $x \in \mathbb{R}$. Clearly indicate any asymptotes and axis intercepts.

(b) Write down the equation of the vertical asymptote of the graph of $g(x)$.

(c) Describe a single transformation that maps the graph of $f(x)$ onto the graph of $g(x)$.

(d) Use your sketch to find the time, $x$, when the populations of culture A and culture B are equal.