Exercises: Logarithms

Calculate the value of the following logarithms by directly applying the definition:

• $\log_2 64$

• $\log_3 \frac{1}{9}$

• $\log_{10} 0,001$

• $\log_5 \sqrt{5}$

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

• $\log_x 81 = 4$

• $\log_2 x = 5$

• $\log_7 x = -2$

Knowing that $\log 2 \approx 0,301$ and $\log 3 \approx 0,477$, calculate the approximate numerical value of $\log 6$ and $\log 1,5$ using the properties of logarithms.

Simplify the following mathematical expression to a single logarithm and calculate its numerical value:

$\log 5 + \log 20 - \log 10$

Calculate the value of $x$ in the exponential equation $3^x = 45$. Express the final result rounded to two decimal places (you may use a calculator for the final step).

Apply the properties of logarithms in reverse to express the following combination as a single logarithm:

$3 \log x + \frac{1}{2} \log y$

The pH of a chemical solution is defined by the formula $pH = -\log [H^+]$, where $[H^+]$ represents the concentration of hydrogen ions measured in mol/L. If a sample of lemon juice has a hydrogen ion concentration of $10^{-3}$ mol/L, what is its pH value?

Use the change-of-base formula to calculate the value of $\log_5 120$ using only the common logarithm ($\log$) or natural logarithm ($\ln$) key on your calculator. Round the result to three decimal places.

Without using a calculator in the first step, bound the value of $\log_3 50$ between two consecutive integers, justifying your answer based on powers of base $3$. Next, use the change of base formula and the calculator to find its approximate value rounded to two decimal places. Finally, check whether the numerical result is consistent with your initial estimate and reason whether this value is closer to the lower or the upper bound of the interval you deduced.

Lucía tried to simplify the expression $\log(2x + 8)$ on her exam and wrote the following steps:

1. a) Point out the conceptual error made by Lucía and explain in detail why her first step is mathematically false, based on the properties of logarithms.
2. b) Show that the equality is incorrect by substituting $x = 1$ and calculating the value of the original expression and the value obtained by Lucía.
3. c) If we wanted to transform $\log(2x + 8)$ in a valid way so that a sum of logarithms appears, how should we manipulate the argument beforehand by factoring?

Solve the logarithmic equation $\log_3 (2x + 5) = 2$ by applying the definition of a logarithm to rearrange for the unknown $x$.

Expand the following logarithmic expression by applying all possible properties, so that it is expressed in terms of $\log x$, $\log y$ and $\log z$:

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

Combine the following algebraic expression into a single logarithm, applying the corresponding properties:

$2 \log a - 3 \log b + \log c - \frac{1}{3} \log d$

A principal of $5000$ € is deposited in a bank at a compound interest rate of $3,5\,\%$ per year. How many full years must elapse for the accumulated amount to exceed $7000$ €?

Remember that the formula for the final amount is $C_f = C_i \cdot \left(1 + \frac{r}{100}\right)^t$.

A population of bacteria doubles every hour. If there are initially $1500$ bacteria, the number of bacteria after $t$ hours is given by the function $P(t) = 1500 \cdot 2^t$. How much exact time must pass for the population to reach $100000$ bacteria?

The magnitude of an earthquake on the Richter scale is calculated using the formula $M = \log \left( \frac{A}{A_0} \right)$, where $A$ is the maximum amplitude of the seismic waves and $A_0$ is a standard reference amplitude. If a particular earthquake records an amplitude $1000$ times greater than $A_0$, what is its magnitude on the Richter scale? Furthermore, determine how many times greater the amplitude of an earthquake of magnitude $6$ is compared to one of magnitude $4$.

Knowing only that $\log 2 \approx 0,301$, calculate the approximate value of $\log 25$ and $\log \sqrt[3]{0,02}$ without entering the direct operation into the calculator, relying only on the properties of logarithms.

A new vehicle has an initial cost of $24000$ €. Due to market wear and tear, its value depreciates by $12\,\%$ each year. The function describing its residual value after $t$ years is $V(t) = 24000 \cdot (0,88)^t$. How many years must pass for the value of the car to be reduced to exactly half of its original price?

The graph of a logarithmic function of the form $f(x) = \log_a (x + b)$ passes through the origin $(0,0)$ and through the point $(6, 1)$.

1. a) Determine, with reasoning, the exact values of the parameters $a$ and $b$.
2. b) From the values found, state the domain of definition of the function and the equation of its vertical asymptote.
3. c) Set up and solve the equation needed to calculate the value of $x$ that satisfies $f(x) = 3$.

The noise pollution protection regulations of a Spanish municipality set $30\text{ dB}$ as the maximum permitted limit inside a bedroom during nighttime hours. The relationship between the sound pressure level $L$ (in decibels, $\text{dB}$) and the physical sound intensity $I$ (in $\text{W/m}^2$) is given by:

where $I_0 = 10^{-12}\text{ W/m}^2$ is the human hearing threshold.

1. a) Calculate the physical intensity $I$ that corresponds exactly to the legal limit of $30\text{ dB}$.
2. b) During emergency roadworks on the street, $50\text{ dB}$ are measured in the bedroom. A neighbor claims that the sound is "less than twice as intense" because $50$ is not quite twice $30$. Refute their claim mathematically by calculating how many times more intense the sound wave from the roadworks is compared to the legal limit in the regulations.

Solve the exponential equation $5^{x+1} = 2^{3x-2}$ by taking common logarithms of both sides and rearranging for the unknown $x$. Express the final result rounded to three decimal places.

The sound intensity level, measured in decibels (dB), is calculated using the physics formula $L = 10 \cdot \log \left( \frac{I}{I_0} \right)$, where $I$ is the sound intensity in $\text{W/m}^2$ and $I_0 = 10^{-12} \text{ W/m}^2$ represents the minimum threshold of human hearing. If the heavy traffic noise on an avenue records a level of $85$ dB, what is its sound intensity $I$? If a jackhammer on a construction site emits twice the sound intensity of that traffic, how many exact decibels will the sound level meter record?

The amount of Carbon-14 ($C-14$) in an organic archaeological remain follows the law of radioactive decay $N(t) = N_0 \cdot e^{-\lambda t}$, where the decay constant is $\lambda \approx 0,0001209$ and $t$ is the elapsed time in years. If laboratory analysis detects that only $30\,\%$ of its original amount of $C-14$ remains in a fragment of prehistoric bone (that is, $N(t) = 0,30 \cdot N_0$), what is the estimated age of the bone?

Simplify the following combined logarithmic expression and calculate its exact numerical value without using a calculator:

$\frac{\log_2 8 + \log_3 81 - \log 1000}{\log_5 125 \cdot \log_4 \sqrt{4}}$

On a well-known social network, the number of followers of an account with viral content grows according to an exponential model $S(t) = S_0 \cdot k^t$, where $t$ is measured in days. It is known that on the initial day ($t = 0$) the account had $200$ followers and on day $5$ it reached $6400$ followers. First, calculate the exact value of the growth constant $k$. Then, determine on which exact day it will pass the one-million-follower mark.

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

Prove, using the change-of-base formula and the fundamental properties of logarithms, that $\log_a b \cdot \log_b a = 1$ for any numbers $a, b > 0$ and different from $1$. Once proven, apply this result to mentally calculate the value of $\log_8 2$, knowing beforehand that $\log_2 8 = 3$.

The following graph represents the population growth of two bacterial colonies, A and B, in a laboratory. The scale of the vertical axis (which indicates the number of bacteria $N(t)$) is logarithmic in base 10, while the horizontal axis (time in hours) has a linear scale. By observing the line describing the behaviour of colony A, find its associated exponential function $N(t) = N_0 \cdot 10^{k \cdot t}$, knowing that it passes exactly through the points $(0, 10^2)$ and $(4, 10^4)$. At what time $t$ will colony A reach exactly one million bacteria ($10^6$)?

To solve the equation $\log_2 x + \log_4 x = 6$, two students propose different strategies:

• Alejandro's method: Express $\log_4 x$ in base $2$ using the change of base formula and solve for $\log_2 x$.

• Berta's method: Transform the first term using the property $\log_2 x = \log_4 (x^2)$ to directly add the logarithms in base $4$ using the logarithm of a product.

1. a) Fully develop the solution following Alejandro's method.
2. b) Formally prove that the equivalence proposed by Berta, $\log_2 x = \log_4 (x^2)$, holds true for all $x > 0$ using the change of base, and finish solving the equation with her procedure, checking that you reach the same solution.
3. c) Compare both approaches: which of the two methods do you consider safer against potential domain errors or algebraic manipulation mistakes? Justify your answer.

The National Geographic Institute (IGN) recorded a seismic sequence in southern Spain in which two earth tremors stood out: the main earthquake $T_1$, with magnitude $M_1 = 4{,}4$, and a later aftershock $T_2$, with magnitude $M_2 = 3{,}1$. The magnitude on the Richter scale is defined as $M = \log\left(\frac{A}{A_0}\right)$, where $A$ is the maximum amplitude recorded by the seismograph and $A_0$ is a calibration amplitude.

1. a) Prove, by applying logarithm properties, that the ratio between the amplitudes of both earthquakes is given by $\frac{A_1}{A_2} = 10^{M_1 - M_2}$, without needing to know the value of $A_0$.
2. b) Calculate how many times greater the ground motion amplitude of $T_1$ was compared to that of $T_2$.
3. c) A third earthquake $T_3$ produced a wave amplitude exactly three times the amplitude of $T_1$ ($A_3 = 3 \cdot A_1$). Knowing that $\log 3 \approx 0{,}477$, calculate the exact magnitude $M_3$ of this third tremor and explain why tripling the physical amplitude of the earthquake is by no means equivalent to tripling its magnitude on the Richter scale.

Unit Challenge. How is the energy of an earthquake measured? Analyse the non-linear relationship between magnitude and energy using official data.

1. Data search: Access the IGN earthquake catalogue (ign.es) and write down three recent earthquakes in Spain with approximate magnitudes $M_1 \approx 2,0$, $M_2 \approx 3,5$, and $M_3 \ge 4,5$.

2. Algebra and notation: The energy released (in joules) follows $E = 10^{4,8} \cdot 10^{1,5M}$. Express $10^{1,5M}$ as a radical using powers with rational exponents ($\sqrt{10^{3M}}$). Use the properties of powers to calculate $E$ in scientific notation for your three values and determine how many times more energetic $M_3$ is than $M_1$.

3. Digital modelling: Solve for $M$ to obtain the logarithmic function $M(E) = \frac{2}{3}(\log E - 4,8)$. Plot this function and its inverse (radical or fractional power) in GeoGebra.

4. Final report: Write a brief document (one page) that includes the IGN data table, the calculations, the graph, and an explanation of why decimal increases in magnitude drastically multiply destructive energy.

Evaluation criteria:

• Rigorous collection of real IGN data and correct calculation of energy in scientific notation.

• Accurate algebraic manipulation between integer exponents, rational exponents, radicals, and logarithms.

• Appropriate plotting and graphical interpretation of the model in GeoGebra.