Logarithmic Function

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 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:

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

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 $g(x) = \ln(x+2) - 3$.

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 $V(t) = 90\,000 \times e^{-0.25t}$, where $V$ is the value in pesos 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 $(g \circ f)(x)$.

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:

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

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 $(2, 0)$.

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

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

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

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

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 liter. 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 $5.0$.

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 $35^\circ\text{C}$.

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.