Operations on algorithms - the logarithm of the product and the logarithm of the power - Operations on algorithms - the logarithm of the product and the logarithm of the power

R1SI3r42jhkfA

Działania na logarytmach – logarytm iloczynu i logarytm potęgi

Learning objectives

You learn the formulas for the logarithm of the product and the logarithm of power. You use formulas for the logarithm of the product and the logarithm of power.

Learning effect

You learn the formulas for the logarithm of the product and the logarithm of power.
You use the formulas for the logarithm of the product and the logarithm of power.

R12qZ4xnwWyYM1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Think what you remember about logarithms and operations on the powers. The aim of the lesson is to learn and use in the calculations formulas for the logarithm of the product and the logarithm of power.

R1LNH7N5ipSLJ1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Analyze the Slideshow showing the method of obtaining the formula for the logarithm of the product and the logarithm of power. Write down the appropriate theorems.

R12CGsuJx9l0R1

Theorem - the logarithm of the productlogarithm of the productlogarithm of the product

R1btXz3uBswbQ1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

With a positive and different from 1 base $a$ , for any number $x > 0$ and $y > 0$ a formula occurs.

\log_{a} (x \cdot y) = \log_{a} x + \log_{a} y

Theorem - the logarithm of powerlogarithm of the powerlogarithm of power

R6oaRiylLsMKB1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

With a positive and different from 1 basis $a$ for any number $x > 0$ and any rational number $r$ there is a formula.

\log_{a} x^{r} = r \cdot \log_{a} x

Using the formulas, solve the following problems.

Task 1

Calculate the value of the terms:

$\log_{6} 2 + \log_{6} 12 + \log_{6} 9$ ,
$\log 4 + \log 25$ ,
$\log_{15} 9 + \log_{15} 25$ ,
$\log_{\frac{1}{8}} 2 + \log_{\frac{1}{8}} 4$ .

Task 2

Calculate the value of the terms:

$\log_{3} 729$ ,
$\log_{2} \frac{1}{32}$
$\log_{6} 27 + \log_{6} 8$ ,
$2 \log_{12} 2 + 2 \log_{12} 6$ .

Task 3

Calculate the value of the terms if $\log_{5} 2 \approx 0, 43$ , $\log_{5} 7 \approx 1, 21$ :

$\log_{5} 14$ ,
$\log_{5} 56$ ,
$\log_{5} 70$ ,
$\log_{5} 3,5$ .

Task 4

Calculate $x$ :

$\log_{5} x = 2 \log_{5} 4$ ,
$\log_{2} x = - 3 \log_{2} \frac{4}{3}$ ,
$\log_{7} x = 2 \log_{2} \frac{3}{4}$ ,
$\log_{\frac{1}{2}} x = - 2 \log_{\frac{1}{2}} 7$ .

Task 5

Write in the form of a logarithmlogarithmlogarithm:

$1 + \log_{2} 5 - \log_{2} 10$ ,
$2 \log 3 + \log 2 + 1$ ,
$\frac{1}{3} \log_{3} 2 + 2$ ,
$\log_{4} 6 + 2$ .

Task 6

R1CQ7MVFdY4K01

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Prove that for any positive real numbers $x$ and $y$ the given equality is true:

$\log \frac{x}{y} + \log \frac{y}{x} = 0$ ,
$\log_{2} x^{2} y - \log_{2} x y^{2} = \log_{2} x - \log_{2} y$ .

Task 7

An extra task:

$\log_{6} x = 2 \log_{6} 4 - \log_{6} 18$ ,
$\log_{3} x = - 2 + 2 \log_{3} 2$ .

Remember:

With a positive and different from 1 base $a$ , for any number $x > 0$ and $y > 0$ a formula occurs.
$\log_{a} (x \cdot y) = \log_{a} x + \log_{a} y$
With a positive and different from 1 basis $a$ for any number $x > 0$ and any rational number $r$ there is a formula.
$\log_{a} x^{r} = r \cdot \log_{a} x$

Do the revision exercises.

Exercises

Exercise 1

R1EjP8GGfSwuG

Which sentence is true? $\log 12 = ...$

$\log 3 \cdot \log 4$
$\log 3 + \log 4$
$\frac{\log 24}{\log 2}$

Rzadanie

Source: GroMar, licencja: CC BY 3.0.

Exercise 2

Prove that $\log_{2} \sqrt{6} + \log_{2} \sqrt{8} - \log_{2} \sqrt{3}$ is a rational number.

Exercise 3

Describe in English how to calculate the following term $\log_{\frac{1}{3}} 9 + 2 \log_{\frac{1}{3}} 27$ .

Exercise 4

R1T7sfRFDBIaJ

Indicate which pairs of expressions or words are translated correctly.

logarytm - logarithm
logarytm iloczynu - logarithm of the product
logarytm potęgi - logarithm of the power
suma logarytmów - sum of logarithms
potęga potęgi - exponent
wykładnik potęgi - base of logarithm

zadanie

Source: GroMar, licencja: CC BY 3.0.

R1VE9nnjrIYja1

Match Polish terms with their English equivalents.

logarithm of the product
logarithm of the power
sum of logarithms
logarytm
logarytm potęgi
suma logarytmów
logarithm
the product of powers of the same base
logarytm iloczynu
iloczyn potęg o tej samej podstawie

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

logarithm

logarytm

R18K3L3mky3KE1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: logarithm

logarithm of the product

logarytm iloczynu

Rn3zNHtAVBIji1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: logarithm of the product

logarithm of the power

logarytm potęgi

RkilvDE5OVqyY1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: logarithm of the power

sum of logarithms

suma logarytmów

RcPLOR4nbJjCf1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: sum of logarithms

product of powers of the same base

iloczyn potęg o tej samej podstawie

RkWemWAqrzebg1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: product of powers of the same base

power of power

potęga potęgi

R1ZC7qxHhe5ut1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: power of power

exponent

wykładnik potęgi

R17XKT0RgNkyA1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: exponent

base of logarithm

podstawa logarytmu

R1LNQiVuQfOAW1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: base of logarithm

Keywords

logarithmlogarithmlogarithm

logarithm of the productlogarithm of the productlogarithm of the product

logarithm of the powerlogarithm of the powerlogarithm of the power

sum of logarithmssum of logarithmssum of logarithms

product of powers of the same baseproduct of powers of the same baseproduct of powers of the same base

Scenariusz

Operations on algorithms - the logarithm of the product and the logarithm of the power

Exercises

Glossary

Keywords