Exercises: Exponential and logarithmic functions

Without plotting them, state which of these exponential functions are increasing and which are decreasing, justifying your answer based on the base:

1. a) $f(x) = 3^{x}$

1. b) $g(x) = 0,5^{x}$

1. c) $h(x) = e^{x}$

1. d) $i(x) = \left(\frac{1}{4}\right)^{x}$

1. e) $j(x) = 1,03^{x}$

Every exponential function $f(x) = a^{x}$ (with $a > 0$ and $a \neq 1$) passes through a common point, and every logarithmic function $g(x) = \log_{a} x$ passes through another. What are these points? Prove it by calculating $f(0)$ and $g(1)$ for any base.

State the equation of the asymptote of each function and specify its type (horizontal or vertical):

1. a) $f(x) = 2^{x}$

1. b) $g(x) = 5^{-x}$

1. c) $h(x) = \log_{3} x$

1. d) $i(x) = \ln x$

Let $f(x) = 3^{x}$ and $g(x) = \log_{2} x$. Calculate, without a calculator:

1. a) $f(-2)$, $f(0)$ and $f(3)$

1. b) $g(8)$, $g(1)$ and $g\left(\frac{1}{2}\right)$

Match each phenomenon with the type of exponential model that best describes it, increasing (base greater than $1$) or decreasing (base between $0$ and $1$), and justify your choice:

• The cooling of a cup of coffee until it reaches room temperature.

• The balance of a savings account with compound interest.

• The remaining radioactivity of a medical source over the years.

• The number of people reached by a chain of kindness in which each person notifies three friends.

Calculate the domain of the following functions:

1. a) $f(x) = \log(x - 3)$

1. b) $g(x) = 2^{x - 1}$

1. c) $h(x) = \ln(10 - x)$

1. d) $i(x) = e^{x} + 1$

State the range of the following functions, reasoning from the graphs of $f(x) = 2^{x}$ and $g(x) = \log x$:

1. a) $f(x) = 2^{x}$

1. b) $g(x) = 2^{x} - 1$

1. c) $h(x) = \log x$

1. d) $i(x) = \log x + 2$

The graph of an exponential function $f(x) = a^{x}$ passes through the point $(2, 9)$.

1. a) What is the value of the base $a$?

1. b) With the base found, calculate $f(3)$ and $f(-1)$.

1. c) Which point does the graph pass through regardless of the value of $a$?

Marcos is working on two tasks about functions and makes common conceptual mistakes:

1. a) To find the domain of the function $f(x) = \log_3(x - 5)$, Marcos states: "Since in fractions you cannot divide by zero, here the only forbidden number is the one that makes the parentheses zero, that is, $x - 5 = 0 \implies x = 5$; therefore, $\text{Dom}(f) = \mathbb{R} \setminus \{5\}$". Explain with reasoning why Marcos's argument is incorrect, what condition a logarithm actually requires, and what the correct domain of $f$ is.

1. b) To solve the equation $4^x = -2$, Marcos writes: "I move the $4$ to the other side dividing and get $x = \dfrac{-2}{4} = -0,5$". Explain why this algebraic step makes no mathematical sense and justify, based on the range of the exponential function, how many real solutions this equation has.

Solve the following questions by constructing the functions from their geometric properties:

1. a) Write the formula of an exponential function of the form $f(x) = a^x$ knowing that it is decreasing and that its graph passes through the point $(-2, 9)$. Justify your choice for the value of the base $a$.

1. b) Write the analytical expression of the logarithmic function $g(x) = \log_b x$ which is the inverse of the function $f(x)$ you obtained in the previous section. What point in the plane does the graph of $g(x)$ necessarily pass through without needing to do any calculations?

1. c) State with reasoning the equation of the asymptote of $f(x)$ and that of $g(x)$, specifying in each case whether it is a horizontal or vertical asymptote.

Consider the exponential function $f(x) = 2^{x}$.

1. a) Copy and complete the table of values of the function for $x \in \{-2, -1, 0, 1, 2, 3\}$.

1. b) What are the domain and the range of the function? Is it increasing or decreasing?

1. c) Which line is its asymptote? Does the graph touch it at any point? Justify your answer.

The figure shows the graphs of $f(x) = 3^{x}$ and $g(x) = \left(\frac{1}{3}\right)^{x}$.

1. a) At what point do the two graphs intersect? Justify that every exponential function of the form $h(x) = a^{x}$ passes through that same point.

1. b) With respect to which axis are the two graphs symmetric? Check it by calculating $f(2)$ and $g(-2)$.

1. c) Without graphing it, describe the graph of $k(x) = 0,7^{x}$: growth, asymptote and position relative to the previous two.

The figure shows the graph of the logarithmic function $f(x) = \log_{2} x$. Each small square of the grid measures 1 unit on both axes.

1. a) State its domain and its range.

1. b) What is its asymptote and what type is it?

1. c) Calculate $f(16)$ and $f\left(\frac{1}{4}\right)$ using the definition of logarithm and check that they fit the graph.

1. d) Does it make sense to calculate $f(-2)$? Why?

Starting from the graph of $f(x) = 2^{x}$ (in blue), the graph of $g(x) = 2^{x} + 3$ (in green) has been drawn, as shown in the figure.

1. a) Which transformation relates both graphs?

1. b) What is the horizontal asymptote of $g$? And its range?

1. c) Write the expression of the function whose graph is that of $f$ shifted $2$ units to the left, and state its asymptote.

The value of a car follows the model $V(t) = 24.000 \cdot 0,85^{t}$, with $t$ in years.

1. a) What was the price of the car when new?

1. b) Calculate its value after $5$ years, rounding to the nearest euro.

1. c) Does the car lose the same value each year? Calculate how much it loses during the first year and how much it loses between the fourth and fifth year, and explain the difference.

The graph shows the percentage of carbon-14 remaining in a sample over time.

1. a) What percentage remains after $5.730$ years? What is this period called?

1. b) Read from the graph the percentage remaining after $17.190$ years (marked with '?') and check it by calculating how many half-lives have passed.

1. c) Determine between which two marks on the time axis the moment when less than $1 \%$ remains will lie, and justify your answer.

The figure shows the graph of $f(x) = \log_{2}(x + 2)$.

1. a) What is the domain of $f$? Relate it to the shaded region and to the asymptote of the graph.

1. b) What is its range?

1. c) Determine, without plotting them, the domain of these functions: $g(x) = \log_{2}(x - 5)$, $h(x) = \ln(4 - x)$, $i(x) = \log(2x + 8)$.

Determine the domain of the following functions:

1. a) $f(x) = 2^{x - 1}$

1. b) $g(x) = e^{\frac{1}{x}}$

1. c) $h(x) = \ln(x^{2} - 4)$

1. d) $i(x) = \log_{3} x + \sqrt{x + 1}$

1. e) $j(x) = \frac{1}{1 - 2^{x}}$

A pension plan marketed in Spain offers an estimated average return of $7\,\%$ compounded annually. If an initial contribution of $2.500$ € is made, the accumulated capital after $t$ years is given by the function $C(t) = 2.500 \cdot 1,07^t$. You want to find out how many years must elapse to double the invested capital (reaching $5.000$ €).

1. a) Before calculating with logarithms, make an estimate: knowing that $1,07^2 \approx 1,14$ and that $1,07^{10} \approx 1,97$, bound the required time between two consecutive whole numbers of years. Justify your estimate.

1. b) Set up the corresponding equation and solve it step by step using logarithms to determine the exact value of $t$ (round the result to the nearest tenth).

1. c) Check whether the result obtained confirms your initial estimate and assess whether it is consistent with the financial Rule of 72 (which predicts that the doubling time at $r\,\%$ annually is approximately $\dfrac{72}{r}$).

According to data from the National Statistics Institute (INE), a rural municipality in Castilla y León had $1.200$ registered residents on January 1, 2015. Due to youth exodus and population aging, its census is decreasing at a rate of approximately $3\,\%$ per year, following the model $P(t) = 1.200 \cdot 0,97^t$, where $t$ is the time in years elapsed since 2015.

1. a) Interpret the meaning of the coefficient $1.200$ and the base $0,97$ in the context of the problem. Justify from the base why the function is strictly decreasing.

1. b) The administration considers that a municipality enters severe risk of depopulation if it falls below $800$ residents. If the trend continues, in which calendar year will this municipality reach this critical threshold? Set up and solve the corresponding equation using logarithms.

1. c) A local headline claims: "At this constant rate of loss of $3\,\%$ each year, in 34 years the town will have lost $102\,\%$ of its residents and will disappear completely". Explain the mathematical fallacy of this claim based on the properties of exponential decay and horizontal asymptotes.

Carbon-14 decays such that the amount remaining in a sample follows the model $N(t) = N_0 \cdot 0,5^{t/5.730}$, where $t$ is the time in years and $N_0$ is the initial amount. An archaeology team analyses a bone fragment found in a cave.

1. a) What percentage of carbon-14 will remain after $2.000$ years? Round to one decimal place.

1. b) If the fragment retains $38 \%$ of the original carbon-14, set up the equation to find its age and solve it using logarithms.

1. c) A newspaper headline states: 'after $11.460$ years there is no carbon-14 left, because two half-lives have passed'. Explain why this is false and calculate the actual percentage remaining at that moment.

The concentration of a medicine in the blood, in mg/l, evolves according to the model $C(t) = 120 \cdot 0,82^{t}$, where $t$ is the time in hours elapsed since its administration.

1. a) What was the concentration at the time of the injection? What about after $6$ hours? Round to one decimal place.

1. b) The drug stops being effective when the concentration falls below $30$ mg/l. Set up the corresponding inequality and calculate, using logarithms, how many hours the effect lasts.

1. c) If a patient receives an identical second dose $12$ hours after the first, what total concentration will they have in their blood immediately after the new injection, assuming the concentrations add up?

The population of a coastal city evolves according to the model $P(t) = 180.000 \cdot 1,015^{t}$, where $t$ is the number of years elapsed since 2025.

1. a) What population is expected in 2035? Round to the nearest hundred.

1. b) In which year will the city exceed $250.000$ inhabitants? Set up the inequality and solve it using logarithms.

1. c) Explain what assumption about growth lies behind this model and reason why it might cease to be realistic in the very long term.

When plotting the functions $f(x) = 2^{x}$ and $g(x) = 2^{-x}$ together, a student claims: 'they are symmetrical with respect to the $X$ axis, because one is the exponential function and the other is its opposite'. Reasonably evaluate the statement: state whether it is true or false, make a sketch of both graphs, explain with respect to which axis they are actually symmetrical and confirm it by comparing $f(2)$ with $g(-2)$.

You can do it on paper and send a photo with your answer.

Two analysts at a music app discuss how the number of users will grow. The first proposes a linear model and the second, an exponential one; both models start from $100$ users in week $0$ and are shown in the figure.

1. a) The line passes through the points $(0, 100)$ and $(10, 1.000)$: write its equation.

1. b) The exponential curve has the form $u(t) = 100 \cdot a^{t}$ and passes through the point $(5, 420)$: calculate the base $a$ using logarithms and round to three decimal places.

1. c) With both complete models, determine between which two consecutive weeks the exponential forecast exceeds the linear one, check it with the figure and explain what difference there is between both forecasts in the long term.

Sound intensity level is measured in decibels (dB) using the logarithmic scale $L = 10 \cdot \log \frac{I}{I_0}$, where $I_0 = 10^{-12}\ \text{W/m}^{2}$ is the threshold of hearing.

1. a) Calculate the level, in dB, of a normal conversation whose intensity is $I = 10^{-6}\ \text{W/m}^{2}$.

1. b) The threshold of pain is at $130$ dB. What is the corresponding intensity? Express it as a power of $10$.

1. c) How many times more intense is the sound of a $110$ dB concert than that of a $40$ dB library? Explain why, although the difference is 'only' $70$ dB, the intensity is multiplied by $10^{7}$.

With an annual inflation rate of $3 \%$, the purchasing power of $1.000\ \text{ €}$ kept 'under the mattress' evolves according to $V(t) = 1.000 \cdot 1,03^{-t}$, where $t$ is the number of years elapsed.

1. a) Calculate the purchasing power after $5$ years, rounded to the nearest cent.

1. b) How many years must pass for that money to be 'worth' half? Set up the equation and solve it using logarithms.

1. c) Roughly plot the function for $t \in [0, 30]$ (suggested scale: $1$ cm $= 5$ years on the horizontal axis and $1$ cm $= 100\ \text{ €}$ on the vertical axis) and mark the results of the previous sections on it.

The value of a new electric car costing $32.000\ \text{ €}$ follows the exponential model $V(t) = 32.000 \cdot 0,85^{t}$, with $t$ in years.

1. a) How much will it be worth after $4$ years? Round to the nearest euro.

1. b) A dealership offers to buy it back when its value falls below $12.000\ \text{ €}$. How much time will pass until then? Use logarithms.

1. c) Another seller uses a linear model according to which the car loses $4.500\ \text{ €}$ of value each year. Compare the value assigned by each model after $4$ years and explain which of the two benefits an owner who wants to sell it in the medium term.

It is proposed to solve in class the exponential equation $4^x - 5 \cdot 2^x + 4 = 0$. Two students suggest different approaches:

• Alba suggests applying logarithms directly to both sides to "bring down" the exponents from the beginning: $\log(4^x - 5 \cdot 2^x + 4) = \log(0)$.

• Borja suggests expressing $4^x$ as $(2^x)^2$ and applying the change of variable $u = 2^x$.

1. a) Evaluate Alba's method: state two rigorous mathematical reasons why her approach is unfeasible (one regarding the domain of the logarithmic function and another regarding the operational properties of the logarithm of a sum or difference).

1. b) Fully develop Borja's method to find the real solutions of the equation, justifying each step when undoing the change of variable using the definition of logarithm.

1. c) If the proposed equation had been $4^x - 5 \cdot 2^x - 6 = 0$, applying the change of variable would yield $u_1 = 6$ and $u_2 = -1$. Explain why the value $u_2 = -1$ must be discarded, connecting your explanation to the range of the exponential function $g(x) = 2^x$.

The attenuation of the light signal in an interurban optical fiber line is modeled by a decreasing exponential function $I(x) = I_0 \cdot a^x$, where $x$ represents the distance traveled by light in kilometers and $I(x)$ is the power of the received signal in milliwatts ($\text{mW}$).

At the transmitting station ($x = 0$), the signal starts with a power of $100\text{ mW}$. Upon reaching a control station located $15\text{ km}$ away, it is detected that the power has decreased to exactly $25\text{ mW}$.

1. a) Deduce with reasoning the exact values of $I_0$ and the base $a$. Express $a$ as a power with an irreducible base (without resorting to decimal approximations) and write the complete analytical expression of $I(x)$.

1. b) The available signal amplifiers require the optical power reaching them to be at least $1,5\text{ mW}$ in order to process the data without loss. Set up and solve using logarithms the maximum distance at which the amplifier can be placed from the transmitting station. Round to the nearest tenth of a kilometer.

1. c) Construct the inverse function $x(I)$ that allows calculating the distance traveled as a function of the observed final intensity. Determine its mathematical domain and the domain with real meaning in the context of this fiber line, analytically justifying what type of function it is.

Topic challenge. How does inflation impact the family economy? You are going to model the evolution of the real cost of living through functions.

1. Data and interpolation: Access the INE website (ine.es) and write down the average annual CPI for the last $5$ years. Apply linear interpolation to estimate the index in an intermediate semester and then fit the data to an exponential model $f(t) = a \cdot b^t$.

2. Inverse and operations: Analytically obtain the inverse function $f^{-1}$ using logarithms to calculate in which year the CPI will exceed a $50\,\%$ increase. Model the fitting error using a piecewise function that integrates the absolute value of the deviation, $E(t) = |f(t) - y_{\text{real}}|$, and analyse whether a square root function $g(t) = k \cdot \sqrt{t}$ would better describe periods of stagnation.

3. Study and final product: Plot the functions in GeoGebra or a spreadsheet. Carry out the complete analysis step by step: domain, range, asymptotes, intercepts with the axes, and sign. Submit a one-page executive report with your graphs and a critical assessment of the results.

Assessment criteria:

• Mathematical precision in linear interpolation and the calculation of logarithmic inverses.

• Rigour in the analytical study (domain, range, sign, piecewise functions, and absolute value).

• Technical quality in the graphical representation with GeoGebra or the spreadsheet.

• Coherence and critical thinking in the socioeconomic conclusions of the report.