Exercises: Logarithms

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.

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) $\log_7 1$

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

Sound intensity is measured in decibels ($dB$) using a logarithmic scale. If the simplified formula to compare two sounds is $dB = 10 \cdot \log_{10}(\frac{I}{I_0})$, 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 $I_0$.

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.

Mateo 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 Mateo 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.

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 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).

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 $[H^{+}] = 2,5 \cdot 10^{-4}$ 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.

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

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

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

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 $M = C \cdot (1+i)^t$.

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.

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

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

$\log E = 1,5 \cdot M + 4,8$

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?

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.

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.

Within the framework of the National Integrated Energy and Climate Plan (PNIEC), an autonomous community generated $1.200\text{ GWh}$ through solar photovoltaic energy in the year 2020. Due to the pace of installation of new solar farms, its production is growing at an estimated rate of $14\%$ per year, modeled by the function $P(t) = 1.200 \cdot (1,14)^t$, where $t$ is the time in years elapsed since 2020 and $P(t)$ is the production in $\text{GWh}$.

1. a) Set up the equation needed to determine how many years must elapse to quadruple the production of 2020 (to reach $4.800\text{ GWh}$) and solve for $t$ by applying common logarithms and their properties.
2. b) Calculate the value of $t$ rounding to two decimal places and interpret in which calendar year and month that target will be reached.
3. c) If the annual growth rate had been twice as high ($28\%$ per year instead of $14\%$), would it take exactly half the time to quadruple production? Justify your answer by reasoning about the properties of logarithms rather than merely giving the numerical result.

A biotechnology laboratory measures the size of a protein as $4,25 \cdot 10^{-9}$ metres with a maximum error of $2\%$. Simultaneously, they measure the distance between two cells as $0,00034$ metres with an absolute error of $1,5 \cdot 10^{-6}$ 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?

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 $b$:

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?

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 $15.000 €$ 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 $4,5\%$?

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

To solve the exponential equation $5 \cdot 2^{3x} = 3 \cdot 4^{x+1}$, two 1st-year Bachillerato students propose completely different strategies:

• Candela decides to apply common logarithms directly to both sides of the equation from the first step: $\log(5 \cdot 2^{3x}) = \log(3 \cdot 4^{x+1})$.

• Sergio prefers to first transform the powers to a common base ($4 = 2^2$), group the exponential terms on one side and the numerical coefficients on the other, and apply logarithms only at the end.

1. a) Fully work out the solution using Candela's method and using Sergio's method, verifying that both lead to the same exact solution expressed as a quotient of common logarithms.
2. b) Analyze both paths: which of the two requires fewer algebraic transformations and minimizes the risk of error when handling parentheses with the unknown variable?
3. c) If in the original equation we replace base $4$ with a $7$, obtaining $5 \cdot 2^{3x} = 3 \cdot 7^{x+1}$, explain with reasoning which of the two methods is still applicable and why.

Spanish regulations on noise pollution (Noise Law 37/2003) establish that the sound intensity level $\beta$ measured in decibels ($\text{dB}$) is related to the physical sound intensity $I$ (in $\text{W/m}^2$) through the formula:

where $I_0 = 10^{-12}\text{ W/m}^2$ represents the human threshold of hearing. In an urban residential zone, the daytime limit is set at $65\text{ dB}$.

1. a) An outdoor restaurant terrace generates a noise level of $62\text{ dB}$. The city council authorizes opening another identical terrace right next to it that emits with the same physical intensity $I$. A neighborhood group claims that the combined noise will be $62 + 62 = 124\text{ dB}$, far exceeding the legal limit. Prove using the properties of logarithms why this direct addition is an error, and calculate the actual combined sound level ($I_{\text{total}} = 2I$). Is the $65\text{ dB}$ limit exceeded?
2. b) Analytically prove that every time the physical intensity of a sound source is multiplied by ten, the sound level in decibels always increases by a fixed and constant amount, and state what this increase is.
3. c) Set up and solve a logarithmic equation to calculate how many terraces identical to the first one would have to operate at the same time for the combined sound level to reach at least $75\text{ dB}$.