Exercises: Logarithms

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

1. a) The number of bacteria after one week is $450,000,000$.
2. b) The size of an individual bacterium is $0.0000000025$ meters.
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 $\log_a b = x \iff a^x = b$:

Exponential Form	Logarithmic Form
$2^5 = 32$	$\log_2 32 = 5$
$3^4 = 81$
$10^{-2} = 0.01$
$5^3 = 125$
$(\frac{1}{2})^3 = \frac{1}{8}$

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) $\log (10 \times k)$
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 $dB = 10 \times \log_{10}(\frac{I}{I_0})$, 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 $I_0$.

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 liter.

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.

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

An investor deposits $100\,000$ pesos 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 $160\,000$ pesos? 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:

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

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?

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.

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

1. a) Calculate the relative error of both measurements in scientific notation.
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 $b$:

$$\log_k \left( \frac{\sqrt[3]{a \times k^2}}{b^3} \right)^2$$

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

$$2\log(x) - \log(x - 16) = 2$$

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:

$$M_w = \frac{2}{3} \log_{10}(E) - 7.9$$

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 $\$300\,000$ 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:

$$\log_{\Phi} (\Phi^n + \Phi^{n+1}) = n + 2$$