Exercises: Exponential functions

Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using roots or fractions if necessary.

Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base:

• $f(x) = 4^x$

• $g(x) = \left(\frac{1}{3}\right)^x$

• $h(x) = 0.8^x$

• $i(x) = e^x$

Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$.

Solve the following exponential equation by equating the bases on both sides of the equation:

$5^{2x-1} = 125$

A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a function of time $t$ (expressed in hours). How many bacteria will be in the culture after $4$ hours?

Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$.

Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0.5^x$, $h(x) = 3^x$ and $i(x) = -2^x$.

Express both sides of the equation as powers with base $2$ and solve:

$8^x = \frac{1}{4}$

Cian has to solve the exponential equation $3 \times 2^{x+1} = 48$. On his exam, he writes the following working:

1. a) Identify and explain in detail the two conceptual errors made by Cian in the first two steps of his solution.
2. b) Solve the equation correctly step by step, explicitly stating which property of powers or exponential functions justifies each transformation.

The appraised value of a utility car bought new in Ireland for $16\,000$ € depreciates by $20\,\%$ each year, such that its value after $t$ years is given by $V(t) = 16\,000 \times 0.8^t$.

1. a) Without setting up formal equations, make a reasoned estimate of between which two consecutive whole years the car's value will drop below half of its original price ($8\,000$ €) for the first time. To justify your estimate, calculate in a simple way and bound the values of $V(1)$, $V(2)$, $V(3)$, and $V(4)$.
2. b) Check your estimate by calculating the exact numerical values of $V(3)$ and $V(4)$ in decimal form. Based on the results, reason whether the car loses the same absolute amount of euros each year or not, and explain how this relates to exponential decay.

Find the algebraic expression for an exponential function of the form $f(x) = k \times a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.

An amount of $5000$ euros is invested in a bank deposit offering an annual compound interest rate of $3\%$.

• Write the function that determines the accumulated capital $C(t)$ after $t$ years.

• Calculate the total capital that will have accumulated after $10$ years.

Solve the following exponential equation by making an appropriate substitution:

$9^x - 10 \times 3^x + 9 = 0$

Determine the domain of definition of the following function:

$f(x) = \frac{x+2}{e^x - e^2}$

A radioactive isotope has a half-life (the time it takes for its initial mass to be reduced by half) of $8$ days. Starting from an initial sample of $200$ grams:

• Write the function $M(t)$ that gives the remaining mass of the isotope as a function of time $t$ in days.

• What mass of this isotope will remain after $24$ days?

Solve the following exponential equation using logarithms:

$4^{x-1} = 15$

Express the result in exact form using logarithms and also provide an approximation rounded to two decimal places.

Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function $g(x) = 2^{x-3} + 1$. Determine the domain, range and the equation of the horizontal asymptote of $g(x)$.

The market value of an industrial machine decreases exponentially over time. It was bought for $40000$ euros and it is known that each year it loses $15\%$ of its value relative to the immediately preceding year.

• Write the function $V(t)$ that gives the value of the machine as a function of the years elapsed.

• What will its value be after $5$ years?

• Exactly how much time must elapse for its value to be halved?

Design an exponential equation that simultaneously satisfies these three conditions:

1. It is solved by making the substitution $t = 2^x$, transforming into a quadratic equation in the variable $t$.

2. Solving the quadratic equation in $t$ yields two distinct real roots, but the original exponential equation in the variable $x$ has a single real solution, which must be exactly $x = 3$.

3. The constant term of the quadratic equation in $t$ must be a strictly negative integer.

1. a) State the equation in $t$ and reason what value the second root must have so that it does not generate any real solution when undoing the substitution.
2. b) Write the complete exponential equation in the variable $x$, expanded in the form $a \times 4^x + b \times 2^x + c = 0$ with $a, b, c \in \mathbb{Z}$.
3. c) Verify analytically that, when solving your exponential equation, the only real solution obtained is $x = 3$.

According to censuses from the Ministry for the Ecological Transition and the Demographic Challenge (MITECO), the Iberian lynx population on the Iberian Peninsula grew from approximately $100$ individuals in the year $2002$ ($t = 0$) to about $1\,600$ individuals in the year $2022$ ($t = 20$). Assuming an exponential growth model of the form $P(t) = P_0 \times a^t$, where $t$ is the time in years elapsed since $2002$:

1. a) Determine the exact values of the constant $P_0$ and the annual growth base $a$. Express the base $a$ as an irreducible power with base $2$.
2. b) From the value of $a$, calculate the average annual percentage growth rate predicted by this model, rounding to the nearest $0.1\,\%$.
3. c) Using only the properties of powers with base $2$ (without resorting to decimal approximations or logarithms), determine in which year the model would predict that the population would reach $6\,400$ lynxes. Explain why a purely exponential model ceases to be biologically realistic if projected over the very long term.

The spread of a virus in an isolated population of $10000$ inhabitants follows a logistic growth model given by the function:

$P(t) = \frac{10000}{1 + 99 e^{-0.5t}}$

where $P(t)$ is the number of infected people and $t$ is the elapsed time in days from the start of the outbreak.

• Calculate the initial number of infected people at time $t = 0$.

• Determine how many people will be infected after $10$ days.

• Calculate the exact time required for half of the population to be infected.

Solve the following system of exponential equations:

Given the function $f(x) = e^{\frac{2x}{x-1}}$, carry out an analytical study to determine:

• Its domain of definition.

• Its axis intercepts.

• The equations of its horizontal and vertical asymptotes, if it has any.

Find the values of the parameters $a$ and $b$ so that the following piecewise function is continuous on all of $\mathbb{R}$:

Solve the following exponential equation by factorising out a common factor after applying index laws:

$2^{x+2} + 2^{x-1} - 2^{x-2} = 34$

According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is the ambient temperature and $T_0$ is the initial temperature of the object. A cup of coffee at $90^\circ\text{C}$ is left in a room kept at a constant temperature of $20^\circ\text{C}$. If after $10$ minutes the temperature of the coffee has dropped to $60^\circ\text{C}$:

• Calculate the exact value of the cooling constant $k$.

• Determine the total time elapsed from the beginning for the coffee to reach a temperature of $30^\circ\text{C}$.

Determine analytically the intervals of constant sign (that is, the intervals where the function is strictly positive and where it is strictly negative) of the function:

$f(x) = (x^2 - 5x + 6)e^{-x}$

The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$.

Determine analytically the coordinates of the point of intersection between the functions $f(x)$ and $g(x)$. Then, calculate the exact area of the triangle whose vertices are this point of intersection and the two points where the curves intersect the line $y = e$.

Consider the following two exponential equations:

Equation 1: $3^{x+1} - 2 \times 3^{x-1} = 21$

Equation 2: $9^x - 4 \times 3^{x+1} + 27 = 0$

You have three possible algebraic strategies available:

• Strategy A: Take logarithms of both sides as a direct first step.

• Strategy B: Factor out the power of base $3$ with the smallest exponent.

• Strategy C: Apply the properties of powers to transform the expression using the substitution $t = 3^x$.

1. a) Justify why Strategy A is unfeasible as a first step in both equations, identifying the typical algebraic error associated with $\log(u + v)$.
2. b) For each equation, make a reasoned choice of the optimal strategy between B and C, pointing out which specific algebraic structure of the equation motivates your choice.
3. c) Solve both equations step by step using in each case the strategy you justified as optimal.

At the Met Éireann station in Mullingar (Co. Westmeath), during a winter night, the outside temperature remains constant at $T_a = -5$ ºC. A thermal container with liquid at an initial temperature $T_0 = 75$ ºC is placed outdoors. Its temperature evolves according to Newton's law of cooling:

where $t$ is the elapsed time in minutes and $a \in (0, 1)$ is the thermal decay base per minute.

1. a) After $15$ minutes of exposure, the temperature of the liquid is $35$ ºC. Calculate the exact value of the base $a$ in the form of an irreducible power and write the final analytical expression for $T(t)$.
2. b) Formally prove that $T(t)$ is a strictly decreasing function for all $t \ge 0$. Find the horizontal asymptote of the function as $t \to +\infty$ and interpret its physical meaning in the context of the experiment.
3. c) A student claims that, if one waits long enough, the thermometer will read exactly $-5$ ºC. Set up the corresponding equation and prove, from the range of the exponential function, why the continuous mathematical model refutes this claim. Briefly explain why this difference arises between the theoretical prediction and the physical reality of thermodynamics.