Exponential and Logarithmic Functions

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 $g\left(\frac{1}{2}\right)$

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 $a$?

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.

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) $j(x) = 1.03^{x}$

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.

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.

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.

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) $i(x) = \ln x$

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?

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) $i(x) = \log x + 2$

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

Carbon-14 decays such that the amount remaining in a sample follows the model $N(t) = N_0 \times 0.5^{t/5,730}$, 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 population of a coastal city evolves according to the model $P(t) = 180,000 \times 1.015^{t}$, 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.

Sound intensity level is measured in decibels (dB) using the logarithmic scale $L = 10 \times \log \frac{I}{I_0}$, 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 $10^{7}$.

The value of a car follows the model $V(t) = 24,000 \times 0.85^{t}$, 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 pesos.

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 concentration of a medicine in the blood, in mg/l, evolves according to the model $C(t) = 120 \times 0.82^{t}$, 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?

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 $u(t) = 100 \times a^{t}$ 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.

With an annual inflation rate of $3 \%$, the purchasing power of $\$20\,000$ kept 'under the mattress' evolves according to $V(t) = 20\,000 \times 1.03^{-t}$, 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 $= \$2\,000$ on the vertical axis) and mark the results of the previous parts on it.

The value of a new electric car costing $\$640\,000$ follows the exponential model $V(t) = 640\,000 \times 0.85^{t}$, with $t$ in years.

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

1. b) A dealership offers to buy it back when its value falls below $\$240\,000$. How much time will pass until then? Use logarithms.

1. c) Another seller uses a linear model according to which the car loses $\$90\,000$ 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.