The logarithmic function

Determine the domain of definition of the logarithmic function $f(x) = \log_3(2x - 8)$.

When calculating the domain and the vertical asymptote of the function $f(x) = \log_5(8 - 2x)$, Cian writes on his exam:

"Step 1: For the logarithm to exist, its argument cannot be negative, so I set $8 - 2x \geq 0 \implies 8 \geq 2x \implies x \leq 4$. The domain is $(-\infty, 4]$."

"Step 2: Since the logarithm vanishes or approaches infinity at the boundary of the domain, the line $x = 4$ is its vertical asymptote, and the curve intersects this asymptote at the point $(4, 0)$."

1. a) Identify the two conceptual errors made by Cian and explain why they are contradictory to each other according to the definition of a vertical asymptote.
2. b) Correct the resolution: write the correct domain in interval notation, calculate the equation of the vertical asymptote with justification, and determine the actual intersection point of the graph with the x-axis.

Determine the domain of definition of the function $f(x) = \ln(x^2 - 4x - 5)$. Express the result using interval notation.

Determine the domain of definition of the function $f(x) = \log_{10} \left( \frac{x+3}{|x-2|} \right)$. Pay special attention to the points that make the numerator, the denominator and the argument of the logarithm equal to zero.

Using the laws of logarithms, expand the following expression to write it as the sum and difference of simpler logarithms:

$\log_5 \left( \frac{25x^3}{y} \right)$

Consider the logarithmic function $f(x) = \log_2(x - 3)$.

1. a) Without using a calculator, estimate between which two consecutive integers the value of $f(25)$ lies. Justify your prediction by bounding the argument between two consecutive powers of base $2$ and argue whether the value will be closer to the smaller or the larger integer.
2. b) Calculate the numerical value of $f(25)$ with a calculator using the change-of-base formula (using common or natural logarithms), round the result to two decimal places, and check if it confirms your estimation.
3. c) A student claims that if $f(x)$ takes a negative value, then $x$ must necessarily be negative. Solve the equation $f(x) = -2$ exactly and use the result to refute this claim, verifying that the solution belongs to the domain of the function.

Given the function $f(x) = \ln \left( \frac{1+x}{1-x} \right)$ whose domain is the interval $x \in (-1, 1)$, prove analytically, using the laws of logarithms, that the identity $f(a) + f(b) = f\left( \frac{a+b}{1+ab} \right)$ holds for any values $a$ and $b$ belonging to the domain that make the expression meaningful.

Find the axis intercepts (x-axis and y-axis, if they exist) of the function $g(x) = \ln(x - 2)$.

Given the function $h(x) = \log_{0,5}(x)$, state with reasoning whether it is an increasing or decreasing function across its entire domain, and calculate the exact value of $h(1)$ and $h(4)$.

Describe precisely what geometric transformations (translations) must be applied to the graph of the basic function $f(x) = \ln(x)$ to obtain the graph of the function $g(x) = \ln(x+2) - 3$.

Study the continuity of the following piecewise function at the point $x = 1$. If any discontinuity exists, classify it by stating its type.

$f(x) = \begin{cases} x^2 - 1 & \text{if } x < 1 \\ \ln(x) & \text{if } x \ge 1 \end{cases}$

Determine the algebraic expression of a logarithmic function of the form $f(x) = \log_a(x + b) + c$ knowing that it simultaneously satisfies the following conditions:

1. Its domain of definition is the interval $(-3, +\infty)$ and the line $x = -3$ is its vertical asymptote.

2. Its graph intersects the x-axis at the point $(5, 0)$.

3. The graph passes through the point $(1, -1)$.

To solve it:

1. a) Reasonably deduce the value of parameter $b$ from the information about the domain and the asymptote.
2. b) Set up a system of equations with conditions 2 and 3 and solve it using the properties of logarithms to find the exact values of the base $a$ and the constant term $c$.
3. c) Justify whether the resulting function is strictly increasing or decreasing on its domain and calculate the coordinates of the intersection point of its graph with the y-axis.

Determine the numerical values of the parameters $a$ and $b$ so that the logarithmic function $f(x) = \ln(ax + b)$ simultaneously satisfies two conditions: having a vertical asymptote at the line with equation $x = -3$ and passing through the point with coordinates $(2, 0)$.

Write the algebraic expression for the inverse function $f^{-1}(x)$ of the exponential function $f(x) = 10^{x+2} - 5$.

From the following graphical representation of the function $f(x) = \log_2(x+2) - 1$, determine analytically its domain, its range, the equation of its vertical asymptote and the exact coordinates of its x- and y-intercepts.

The book value of a high-capacity computer server depreciates over time according to the exponential function $V(t) = 4500 \times e^{-0.25t}$, where $V$ is the value in euros and $t$ is the time elapsed in years since its purchase. Find the algebraic expression for the inverse function $t(V)$ and briefly explain what quantity this new function allows us to calculate.

Analyse the symmetry relationship between the functions $f(x) = e^{x-1}$ and $g(x) = \ln(x) + 1$. Observe the attached graph showing both curves and the line y = x (the bisector of the first quadrant). Prove analytically that they are inverse functions by calculating the compositions $(f \circ g)(x)$ and $(g \circ f)(x)$.

A logarithmic function of the form $f(x) = \log_a(b \times x + c)$, with base $a>0$, $a \neq 1$, and coefficient $b>0$, has as its inverse function the exponential function $f^{-1}(x) = \frac{3^x - 7}{4}$.

1. a) Determine with justification the exact values of the constants $a$, $b$, and $c$. To do this, solve for the independent variable from the expression of $f^{-1}$ using the definition of logarithm and the reciprocity property between inverse functions.
2. b) Rigorously determine the domain and range of $f(x)$, as well as the equation of its vertical asymptote. Explain how these elements correspond to the range and domain of $f^{-1}(x)$ through graphical symmetry with respect to the line $y = x$.
3. c) Obtain the simplified algebraic expressions for the composite functions $(f \circ f^{-1})(x)$ and $(f^{-1} \circ f)(x)$. Justify for which set of real numbers each is valid and explain why the fact that their domains do not coincide does not contradict the definition of inverse functions.

Solve the following logarithmic equation by applying the definition of a logarithm:

$\log_2(3x + 1) = 4$

Solve the following logarithmic equation, checking the validity of the solutions obtained:

$\log(x + 2) + \log(x - 1) = 1$

Find the values of $x$ and $y$ that satisfy the following system of non-linear equations:

$\begin{cases} x - y = 9 \\ \log_3(x) + \log_3(y) = 3 \end{cases}$

Solve the following logarithmic equation. Make sure to check possible solutions in the original equation to rule out extraneous roots:

$2\log_2(x) - \log_2(x-1) = 3$

Solve the following system combining exponential and logarithmic equations, finding the values of $x$ and $y$:

$\begin{cases} 2^x \times 4^y = 64 \\ \ln(x) - \ln(y) = \ln(2) \end{cases}$

To solve the logarithmic equation $\log_2(x + 2) + \log_2(x - 1) = 2$, three working strategies are proposed:

• Method 1 (Grouping by product): Apply the product rule for logarithms to transform the equation into $\log_2((x+2)(x-1)) = 2$, convert to exponential form $(x+2)(x-1) = 2^2$, and solve the resulting quadratic equation.

• Method 2 (Equating arguments): Write the constant term as $2 = \log_2(4)$, combine the left-hand side into a single logarithm, and equate the arguments directly: $(x+2)(x-1) = 4$.

• Method 3 (Isolation and quotient): Isolate one logarithm, $\log_2(x+2) = 2 - \log_2(x-1)$, transform the right-hand side into $\log_2(4) - \log_2(x-1) = \log_2\left(\frac{4}{x-1}\right)$, and equate the arguments: $x+2 = \frac{4}{x-1}$.

1. a) Solve the equation by developing the complete algebraic steps using Method 1 and Method 3.
2. b) In all three methods, the algebraic solution $x = -3$ arises. Explain rigorously why this solution is extraneous (invalid), detailing at what exact step of the algebraic manipulation the domain of validity is expanded relative to the original domain of the equation.
3. c) Compare the three methods, analyzing which one is algebraically safer to avoid introducing calculation errors or zero denominators, and justify which indispensable check must always be performed, regardless of the chosen method.

In chemistry, the pH of a solution is defined by the expression $pH = -\log[H^+]$, where $[H^+]$ represents the hydrogen ion concentration in moles per litre. Calculate the pH of an orange juice whose hydrogen ion concentration is $[H^+] = 10^{-3.2}$.

The sound intensity level $\beta$, measured in decibels (dB), is calculated using the acoustic formula $\beta = 10 \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ is the standard threshold of hearing. If the take-off of a jet aircraft generates a noise level of $140 \text{ dB}$, what is the intensity $I$ of that sound in $\text{W/m}^2$?

On the Richter magnitude scale, the magnitude $M$ of an earthquake is related to the energy released $E$ (in ergs) by the empirical equation $\log(E) = 11.8 + 1.5M$. Calculate how many times greater the energy released by an earthquake of magnitude $7.0$ is compared to the energy released by one of magnitude $5.0$.

In Spain, Law 37/2003 on Noise regulates acoustic pollution levels in cities. The sound pressure level $L$, measured in decibels ($\text{dB}$), is modeled by the logarithmic function:

where $I$ is the physical sound intensity in watts per square meter ($\text{W/m}^2$) and $I_0 = 10^{-12}\text{ W/m}^2$ represents the threshold of human hearing.

During the redevelopment works of an avenue in Valencia, two sources of continuous acoustic emission are measured separately: road traffic, which produces an intensity $I_1 = 10^{-5}\text{ W/m}^2$, and a construction compressor, which produces $I_2 = 10^{-4}\text{ W/m}^2$.

1. a) Calculate the sound level in decibels produced by each source separately, $L(I_1)$ and $L(I_2)$, applying the definition of a logarithm without using a calculator.
2. b) A neighborhood association claims that when both sources operate at the same time, the total sound level will be the sum of the decibels of each one ($L(I_1) + L(I_2)$). Knowing that physical intensities add directly ($I_{\text{total}} = I_1 + I_2$), prove using the properties of logarithms why adding decibels is mathematically incorrect and calculate the actual combined noise level $L(I_{\text{total}})$. (Given: $\log_{10}(1.1) \approx 0.041$).
3. c) If the legal limit allowed for that area during daytime hours is $85\text{ dB}$, reason whether the works comply with the law or violate it.

The radioactive decay law of Carbon-14 is given by the function $N(t) = N_0 \times e^{-\lambda t}$, where $N_0$ is the initial amount of isotope, $N(t)$ is the remaining amount at time $t$ (in years) and $\lambda$ is the decay constant. Knowing that the half-life (time required for the initial amount to be reduced to exactly half) is $5730$ years, calculate the exact value of the constant $\lambda$. Subsequently, determine the approximate age of a bone fossil that retains only $15\%$ of its original Carbon-14.

According to Newton's law of cooling, the temperature $T$ of a body 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 constant ambient temperature and $T_0$ is the initial temperature of the body. A cup of tea at $90^\circ\text{C}$ is left to cool in a room whose temperature is $20^\circ\text{C}$. If after $10$ minutes the temperature of the tea has dropped to $60^\circ\text{C}$, find the exact value of the cooling constant $k$. Next, calculate the total time that must elapse from the start for the temperature to drop to $35^\circ\text{C}$.