Logarithmic Functions

Calculate the value of the following logarithms by directly applying the definition, without using a calculator:

a) $\log_2 32$

b) $\log_3 \frac{1}{9}$

c) $\log 10000$

Find the value of the unknown $x$ in each of the following equalities using the definition of a logarithm:

a) $\log_x 25 = 2$

b) $\log_5 x = 3$

c) $\log_2 64 = x$

Given that, approximately, $\log 2 = 0,301$, calculate the value of $\log 8$ using exclusively the properties of logarithms.

Given that $\log 2 = 0,301$ and $\log 3 = 0,477$, calculate the value of $\log 12$ and $\log 5$ using the properties of logarithms, without directly using the log key on the calculator for those numbers. (Hint: remember that $5 = \frac{10}{2}$).

Expand the following expression by applying the properties of logarithms (logarithm of a product, quotient and power), so that only logarithms of simple variables remain:

$\log \left( \frac{x^3 \cdot y^2}{\sqrt{z}} \right)$

Carry out the reverse process to the previous exercise. Group the following expression into a single logarithm using the corresponding properties:

$2 \log a + \log b - 3 \log c$

Use a calculator and the change of base formula to find the value of $\log_4 15$. Round the result to three decimal places.

Determine the domain of definition of the following logarithmic functions:

a) $f(x) = \log(x - 5)$

b) $g(x) = \ln(2x + 8)$

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

Indicate, reasoning your answer and without needing to draw them, which of the following functions are increasing and which are decreasing:

a) $f(x) = \log_3 x$

b) $g(x) = \log_{0,5} x$

c) $h(x) = \ln x$

Observe the graphs of the logarithmic functions represented in the image above. Match, justifying your answer, Function A and Function B with their respective analytical expressions:

1) $y = \log_2 x$

2) $y = \log_{1/2} x$

The image shows the graph of the exponential function $f(x) = e^x$ and its inverse function $g(x)$.

a) Write the analytical expression of $g(x)$.

b) Explain the geometric meaning of the dotted orange line $y = x$ in relation to both curves.

c) Determine the domain and range of both functions, justifying how they are interchanged between a function and its inverse.

Given the function $f(x) = \log_3(12 - 3x)$:

a) Determine its domain analytically.

b) Indicate the equation of its vertical asymptote.

c) In which direction (left or right) does the graph of this function open with respect to its asymptote?

From the graph of the function $f(x)$ shown above, which corresponds to a translated logarithm of the form $f(x) = \log_a (x + b)$:

a) Indicate the value of $b$ by observing the vertical asymptote.

b) Use the point $B(1, 2)$ to calculate the base $a$ of the logarithm.

c) Write the complete analytical expression of the function.

It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin of coordinates $(0, 0)$ and through the point $(6, 1)$.

Set up a system of equations to find the values of the parameters $a$ and $b$. Once found, write the final analytical expression of the function and indicate its vertical asymptote.

Solve the following simple logarithmic equations:

a) $\log(3x - 1) = \log(x + 7)$

b) $\log_2(x - 4) = 3$

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:

$\begin{cases} x - y = 9 \\ \log x - \log y = 1 \end{cases}$

A capital is invested in a bank fund that offers an annual compound interest. The formula governing the accumulated capital $C$ over the years $t$ is $C = C_0 \cdot 1,04^t$, where $C_0$ is the initial capital. If $2000$ dollars are invested, determine how many full years must pass for the capital to exceed $3000$ dollars. Use logarithms to solve for the unknown.

Carbon-14 dating is used to date ancient organic remains. The proportion of C-14 decreases exponentially with time according to the function $P(t) = 100 \cdot 0,5^{\frac{t}{5730}}$, where $P$ is the percentage of C-14 remaining compared to the original and $t$ is the time elapsed in years.

If a fossil is discovered that retains only $15\%$ of its original Carbon-14, what is the estimated age of the fossil? Express the result in years and round to the nearest whole number.

In chemistry, the acidity level of a solution is measured by its pH, which is defined using a base 10 logarithm with the formula $pH = -\log[H^+]$, where $[H^+]$ is the concentration of hydrogen ions in moles per liter. If a sample of lemon juice has a hydrogen ion concentration of $10^{-3}$ mol/L, calculate its pH.

The Richter scale for measuring the magnitude of earthquakes is logarithmic and is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the amplitude of the seismic wave and $A_0$ is a reference amplitude.

An earthquake in city Alpha had a magnitude of $6$ on the Richter scale, and another in city Beta had a magnitude of $8$. How many times greater was the amplitude of the seismic wave of the Beta earthquake compared to that of Alpha? Prove your reasoning mathematically.

Sound intensity is measured in decibels (dB) using a logarithmic function. The formula is $d = 10 \cdot \log\left(\frac{I}{I_0}\right)$, where $d$ is the decibels, $I$ is the sound intensity and $I_0$ is the threshold intensity of hearing (a constant). Make the intensity $I$ the subject in terms of the decibels $d$ and $I_0$.

In computer science, the execution time $T$ of a certain sorting algorithm depends on the amount of data $n$ to be processed, and follows a function of the type $T(n) = k \cdot n \cdot \log_2 n$, where $k$ is a constant specific to the computer.

If when processing $n = 8$ data items the computer takes $T = 24$ milliseconds:

a) Calculate the value of the constant $k$.

b) How long will the same computer take to process $n = 32$ data items?