Logarithmic Functions

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

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

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

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

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 zero.

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

Find the points of intersection with the coordinate axes (x-axis and y-axis, if they exist) of the function $g(x) = \ln(x - 2)$.

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

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 the points of intersection with the x and y-axes.

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 $(2, 0)$.

Study the continuity of the following piecewise function at the point $x = 1$. If there is any discontinuity, classify it by indicating what type it is.

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

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

The book value of a high-capacity computer server depreciates over time according to the exponential function $V(t) = 4500 \cdot e^{-0,25t}$, where $V$ is the value in dollars and $t$ is the time elapsed in years since its purchase. Find the analytical expression of the inverse function $t(V)$ and briefly explain what quantity this new function allows to be calculated.

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 bisector of the first quadrant. Prove analytically that they are inverse functions by calculating the composition $(f \circ g)(x)$ and $(g \circ f)(x)$.

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

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

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 properties of logarithms, that the identity $f(a) + f(b) = f\left( \frac{a+b}{1+ab} \right)$ is satisfied for any values $a$ and $b$ belonging to the domain that make the expression meaningful.

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

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

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

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

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

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

In chemistry, the pH of a solution is defined by the expression $pH = -\log[H^+]$, where $[H^+]$ represents the concentration of hydrogen ions in moles per liter. Calculate the pH of 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 plane generates a noise level of $140 \text{ dB}$, what is the intensity $I$ of that sound in $\text{W/m}^2$?

On the Richter seismological 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 $5,0$.

The law of radioactive decay of Carbon-14 is given by the function $N(t) = N_0 \cdot e^{-\lambda t}$, where $N_0$ is the initial amount of the isotope, $N(t)$ is the remaining amount at time $t$ (in years) and $\lambda$ is the decay constant. Knowing that the half-life (time needed for the initial amount to halve exactly) 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 a 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 rest 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 how much total time must elapse from the start for the temperature to drop to $35^\circ\text{C}$.