Logarithmic functions

Logarithmic functions > Properties

Overview

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Exercises

Exercise 1

Calculate, using the properties of logarithms:

${}^{10} log (5) + {}^{10} log (20)$

${}^{5} log (100) - {}^{5} log (4)$

$2 \cdot {}^{6} log (3) + {}^{6} log (4)$

${}^{\frac{1}{3}} log (45) - {}^{\frac{1}{3}} log (5)$

Exercise 2

Figure out which of these logarithms are easy to evaluate without a calculator. Give an exact answer for those questions. For the remaining questions, approximate the answer to three decimals. Use the log-button of your calculator to do so.

${}^{5} log (625)$

${}^{2} log (100)$

${}^{7} log (\sqrt{7})$

${}^{8} log (8000)$

${}^{\frac{1}{3}} log (50)$

$log (40) + log (25)$

${}^{\frac{1}{3}} log (0.0003)$

Exercise 3

A radioactive substance decays according to the formula:

$N (t) = N (0) \cdot {0.93}^{t}$

$N$ is the amount in mg and $t$ is the time in years.

What is the half-time of this substance?

A laboratory has a stock of $400$ g of this substance. Calculate, using the half-time, how long it takes for the amount of original material to become less than $50$ g.

Calculate to the month how long it takes for $50$ g of original substance to become less than $10$ g.

Exercise 4

The radioactive isotope calcium-45 has a half-time of $165$ days.

After what time has any quantity of calcium-45 been reduced to $1 / 4$ of its original amount?

After what time has any quantity of calcium-45 been reduced to $1 / 8$ of its original amount?

A laboratory has a stock of $100$ grams of calcium-45. Estimate, using your answers from parts a and b, how long it would take for this stock to contain less than $15$ grams of calcium-45.

Calculate a precise answer to c (in days).

Exercise 5

A quantity is growing exponentially with a growth percentage of $p$ percent.

Show that the doubling time $T$ is given by $T = \frac{log (2)}{log (1 + \frac{p}{100})}$ .

Exercise 6

Solve the following equations algebraically. Give approximate answers, rounded to one decimal.

$10 \cdot 5^{x} = 0.16$

${}^{3} log (x^{2}) = 3$

$log (2 x) - 2 \cdot log (x) = 1$

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