Exercises: Logarithmic functions

Calculate the value of the following logarithms by applying the definition directly, 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 equations using the definition of a logarithm:

a) $\log_x 25 = 2$

b) $\log_5 x = 3$

c) $\log_2 64 = x$

Determine the domain of definition of the following logarithmic functions:

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

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

State, giving reasons for your answer and without sketching 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) $h(x) = \ln x$

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

Look at the graphs of the logarithmic functions shown in the image above. Match, giving reasons for your answer, Function A and Function B with their respective algebraic expressions:

1) $y = \log_2 x$

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

Knowing that, approximately, $\log 2 = 0,301$, calculate the value of $\log 8$ using only the laws of logarithms.

Sofía is studying the function $f(x) = \log_3(x - 2)$ for a 4th-year ESO exam and writes down the following three conclusions on her review sheet:

1. "Since the logarithm of negative numbers and zero does not exist, the domain is $[2, +\infty)$, including $2$ because that is where the curve begins."

2. "Since the base is $a = 3$, which is a number greater than $1$, the function is decreasing and has a horizontal asymptote on the $X$-axis."

3. "The graph intersects the $y$-axis ($Y$-axis) at the point $(0, -2)$."

Identify in detail the conceptual error that Sofía makes in each of her three statements and write the correct mathematical justification. Finally, determine with justification the real domain of $f(x)$, the equation of its vertical asymptote, and the exact coordinates of its $X$-intercept.

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

1. a) Without using a calculator, estimate between which two consecutive integers the value of $f(20)$ lies. Justify your deduction relying on the powers of base $2$ closest to $20$ and on the fact that the function is strictly increasing.
2. b) Knowing that $2^{4{,}32} \approx 19{,}97$ and that $2^{4{,}33} \approx 20{,}11$, verify your previous estimate and bound the value of $f(20)$ to two decimal places.
3. c) Notice that the number $20$ is numerically closer to $16$ than to $32$ ($20 - 16 = 4$, whereas $32 - 20 = 12$). Explain, based on the growth and the concave shape of the logarithm graph, why the value of $f(20)$ is so close to $f(16) = 4$ compared to the distance separating it from $f(32) = 5$.

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

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

Combine the following expression into a single logarithm using the properties of logarithms:

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

Solve the following simple logarithmic equations:

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

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

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

a) Determine its domain algebraically.

b) State the equation of its vertical asymptote.

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

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

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) State 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 full algebraic expression for the function.

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 decibel level, $I$ is the sound intensity and $I_0$ is the threshold of hearing intensity (a constant). Rearrange for the intensity $I$ in terms of the decibels $d$ and $I_0$.

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

A logarithmic function of the form $f(x) = \log_a(x + b)$ has its vertical asymptote at the line $x = -5$ and its graph passes through the point $(11, 4)$.

1. a) Determine with reasoning the value of the horizontal shift $b$ and the value of the base $a$. From them, specify the domain of $f(x)$ and reason whether the function is increasing or decreasing over its entire domain.
2. b) Recall that the exponential function and the logarithmic function are inverse functions whose graphs are symmetric with respect to the line $y = x$. Deduce the algebraic expression of the inverse function $f^{-1}(x)$, state its domain and its range, and find the coordinates of the $Y$-intercept of $f^{-1}(x)$ from the $X$-intercept of $f(x)$.

Spanish regulations on acoustic pollution evaluate the sound pressure level in decibels ($\text{dB}$) using the function:

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

1. a) On a central avenue in Valencia, heavy traffic recorded an intensity of $I_1 = 10^{-4}\text{ W/m}^2$. After implementing traffic-calming measures at $30\text{ km/h}$ and a bike lane, the intensity dropped to $I_2 = 10^{-6}\text{ W/m}^2$. Calculate the noise level in decibels before and after the intervention.
2. b) A neighbourhood spokesperson states: "Since the intensity of the physical wave has been reduced to one hundredth ($100$ times less), the decibel level should also have been divided by $100$, dropping from $80\text{ dB}$ to less than $1\text{ dB}$." Applying the properties of the logarithm of a quotient or of a power, mathematically refute the neighbour's claim with rigor and prove by how many decibels the noise level decreases exactly every time the physical intensity $I$ is divided by $10$.

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:

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

The Richter scale for measuring earthquake magnitude 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? Show your reasoning mathematically.

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

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

If a fossil is discovered that contains 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.

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

a) Write the algebraic expression for $g(x)$.

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

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

It is known that a logarithmic function of the form $f(x) = \log_a(x + b)$ passes through the origin $(0, 0)$ 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 algebraic expression for the function and state its vertical asymptote.

In computer science, the execution time $T$ of a sorting algorithm depends on the amount of data $n$ to process, 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 $n = 8$ items of data the computer takes $T = 24$ milliseconds:

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

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

We want to find out whether the graphs of the functions $f(x) = \log_2 x$ and its inverse function $g(x) = 2^x$ have any point of intersection in the Cartesian plane. Two students propose opposite strategies:

• Algebraic method (proposed by Hugo): Equate both expressions by setting up the equation $\log_2 x = 2^x$, apply the definition of logarithm to write $x = 2^{(2^x)}$, and try to solve algebraically for the value of $x$ using basic operations of powers and roots.

• Geometric method and inverse properties (proposed by Irene): Recall that the graphs of two inverse functions are symmetric with respect to the line $y = x$. Irene claims that if an increasing function intersects its inverse, their common points must necessarily lie on the line of symmetry $y = x$. Therefore, instead of solving $\log_2 x = 2^x$, it suffices to analyze the relative position of $g(x) = 2^x$ with respect to the line $y = x$ by comparing the values of $2^x$ and $x$ for $x > 0$.

1. a) Evaluate both methods: explain why Hugo's purely algebraic attempt is a dead end at the 4th-year ESO level and describe why Irene's strategy is conceptually sound.
2. b) Apply Irene's reasoning: calculate the values of $g(x) = 2^x$ and the line $y = x$ for the test points $x = 0$, $x = 1$, and $x = 2$. From them and from exponential growth, formally justify whether $f(x)$ and $g(x)$ ever intersect in the plane.

The Instituto Geográfico Nacional (IGN) calibrates the response of a seismic sensor using a translated logarithmic function of the type $f(x) = \log_a(x + b)$. The following technical conditions about its behaviour are known:

1. The sensor cannot register magnitudes when $x \le -3$, the line $x = -3$ being a vertical asymptote of the curve.

2. The function is strictly decreasing throughout its entire domain.

3. The graph of its inverse function, $f^{-1}(x)$, contains the point $(-1, 1)$.

1. a) Determine with reasoning the parameters $b$ and $a$. Justify why the base $a$ must be a number strictly strictly between $0$ and $1$, and explicitly formulate the functions $f(x)$ and $f^{-1}(x)$.
2. b) Describe the asymptotic behaviour of $f(x)$: relying on the relationship between logarithms and powers with a base strictly between $(0, 1)$, deduce with reasoning what value $f(x)$ approaches as the variable $x$ approaches $-3$ from the right ($x \to -3^+$).
3. c) Find the exact coordinates of the intercepts of $f(x)$ with the $x$-axis ($X$) and with the $y$-axis ($Y$).