Logarithms

To understand the relationship between powers and logarithms, complete the following table by converting each exponential equality into its corresponding logarithmic form, based on the definition $\log_a b = x \iff a^x = b$:

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

1. a) $\log_2 16$
2. b) $\log_3 27$
3. c) $\log_{10} 1000$
4. d) $\log_5 (\frac{1}{5})$
5. e) $\log_7 1$

Calculate the value of the following expressions using exclusively the definition of a logarithm, without using a calculator:

1. a) $\log_{3}\left(\frac{1}{243}\right)$

1. b) $\log_{\sqrt{2}}(32)$

1. c) $\log_{5}(\sqrt[3]{25})$

Given that $\log k = 0.6$, use the properties of logarithms (logarithm of a product, of a quotient and of a power) to calculate the value of the following expressions:

1. a) $\log (10 \times k)$
2. b) $\log (k^2)$
3. c) $\log (\frac{k}{100})$

Given that the common logarithms of 2 and 3 are approximately $\log 2 \approx 0.3010$ and $\log 3 \approx 0.4771$, use the properties of logarithms to estimate the value of:

1. a) $\log(1,2)$

1. b) $\log(\sqrt{18})$

1. c) $\log(25)$ (Hint: relate 5 to 10 and 2).

Expand the following expressions using the properties of logarithms, so that the final result contains only logarithms of simple variables:

1. a) $\log \left( \frac{100 \times x^2}{\sqrt{y}} \right)$

1. b) $\ln \left( \frac{e^3 \times a}{b^2 \times c} \right)$

Given that $\log_k a = 2$ and $\log_k b = 5$, use the properties of logarithms to find the exact value of the following expression without calculating the value of $k$, $a$ or $b$:

Given the expression for the golden ratio $\Phi = \frac{1 + \sqrt{5}}{2}$, prove using logarithms and exponent properties that the following equality holds:

Without using your calculator at first, estimate between which two consecutive integers the value of $\log_3(70)$ must lie, justifying your bounds from integer powers of base $3$. Then, use the change of base formula to common (base-10) logarithms to calculate its numerical value with your calculator (rounding to three decimal places) and verify whether your estimate was accurate by raising base $3$ to the obtained exponent.

Use the change of base formula and your calculator to answer the following questions:

1. a) Calculate the value of $\log_{2}(15) + \log_{5}(2)$ to four decimal places.

1. b) Solve the exponential equation $4^{x+1} = 125$. Round the result to two decimal places.

Cian is solving the equation $\log(x + 20) - \log(x) = 1$ on the board. In his notebook, he writes the following steps:

Step 1: $\log(x + 20 - x) = 1$

Step 2: $\log(20) = 1$

Step 3: "Since $\log(20) \ne 1$, the equation has no real solution".

1. a) Explain precisely what conceptual error Cian made in Step 1 when applying the properties of logarithms, and provide a simple numerical counterexample showing that $\log(A - B) \ne \log A - \log B$.
2. b) Solve the equation correctly by applying the legitimate properties of logarithms, and verify that the obtained solution is valid by substituting it into the original equation.

Solve the following logarithmic equations applying the properties to combine the terms and check the validity of the solutions obtained:

1. a) $2 \log(x) - \log(x + 6) = 0$

1. b) $\log(x+3) + \log(x) = 1$

Design a logarithmic equation of the form $\log(x) + \log(x - k) = 1$ (where $k$ is a positive constant integer) that simultaneously satisfies these two conditions:

1. When transformed into a polynomial equation using the properties of logarithms, it admits the values $x = 5$ and $x = -2$ as candidate solutions.

2. When checking validity in the original equation, only $x = 5$ is a valid real solution, while $x = -2$ must necessarily be discarded.

1. a) Algebraically deduce the exact numerical value that the constant $k$ must have.
2. b) Solve the equation step by step with the obtained value of $k$ and justify rigorously, using the formal definition of a logarithm, why $x = -2$ is rejected and $x = 5$ is accepted.

Solve the following logarithmic equation, ensuring you check the validity of the solutions obtained in the domain of definition:

Contextualise the problem: If $x$ represented the number of users in thousands of a new social network, would both mathematical solutions make sense in the real world?

A surveyor measures the length of a plot of land intended for a new central park and obtains a value of $154.7$ metres. If the exact actual measurement is $154.728$ metres, calculate:

1. a) The absolute error made in the measurement.
2. b) The relative error made, expressed as a percentage and rounded to two decimal places.

In a biology laboratory, it is estimated that the number of bacteria in a sample grows very quickly. Express the following data using standard form:

1. a) The number of bacteria after one week is $450,000,000$.
2. b) The size of an individual bacterium is $0.0000000025$ metres.
3. c) The mass of the processed sample is $0.00034$ grams.

Sound intensity is measured in decibels ($dB$) using a logarithmic scale. If the simplified formula to compare two sounds is $dB = 10 \times \log_{10}(\frac{I}{I_0})$, where $I_0$ is the threshold of hearing, calculate how many decibels correspond to:

1. a) A sound whose intensity $I$ is $100$ times greater than $I_0$.
2. b) A sound whose intensity $I$ is $10,000$ times greater than $I_0$.

In the chemistry laboratory, the acidity level of a substance is measured using $pH$, which is defined as $pH = -\log[H^{+}]$, where $[H^{+}]$ is the hydrogen ion concentration in moles per litre.

If a soft drink has a concentration $[H^{+}] = 2.5 \times 10^{-4}$ and a mineral water has a $pH = 6.2$:

1. a) Calculate the $pH$ of the soft drink.
2. b) Determine the hydrogen ion concentration $[H^{+}]$ of the mineral water.

An investor deposits $5,000$ € in an investment fund that offers compound interest at $3.5 \%$ per annum. The final amount $M$ after a time $t$ in years is given by the formula $M = C \times (1+i)^t$.

How many years must elapse for the investor to reach a final amount of $8,000$ €? Express the result rounded to the nearest whole number.

The magnitude $M$ of an earthquake on the Richter scale is related to the released energy $E$ (in joules) by the formula:

$\log E = 1.5 \times M + 4.8$

The figure shows the graph of this relationship (magnitude $M$ on the horizontal axis and $\log E$ on the vertical axis). Calculate using the formula; the graph only serves to check that the result is reasonable.

1. a) If an earthquake has magnitude $M = 5$, how much energy does it release?

1. b) If an earthquake releases $10^{15}$ joules of energy, what is its magnitude on the Richter scale?

A biotechnology laboratory measures the size of a protein as $4.25 \times 10^{-9}$ metres with a maximum error of $2\%$. Simultaneously, they measure the distance between two cells as $0.00034$ metres with an absolute error of $1.5 \times 10^{-6}$ metres.

1. a) Calculate the relative error of both measurements in standard form.
2. b) Determine with reasoning which of the two measurements is more precise.
3. c) If both lengths were added together, what would be the bound on the absolute error of the sum?

The moment magnitude scale ($M_w$), used to measure the energy of earthquakes, is defined as:

Where $E$ is the energy released in ergs.

The figure shows, schematically, how $\log_{10}(E)$ grows with the magnitude $M_w$ and marks the two earthquakes from the parts below.

1. a) If an earthquake of magnitude $6.5$ is considered "strong", calculate the energy $E_1$ it releases, in scientific notation.

1. b) How many times more energy does an earthquake of magnitude $8.5$ ($E_2$) release compared with one of magnitude $6.5$ ($E_1$)? Express the ratio as a power of base 10.

An investor places $\text{€}\ 15,000$ in a financial product offering a nominal compound interest rate of $4.5\%$ per annum with monthly compounding.

1. a) Using logarithms, calculate exactly how many months must elapse for the final capital to be triple the initial capital.
2. b) Calculate the Annual Equivalent Rate (AER) of this transaction and express it as a percentage to two decimal places. What does this value mean with respect to the nominal $4.5\%$?