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

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

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 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).

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

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?

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.

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?

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 $4,5\%$?