Topicm71c5f897fa3066bd_1528449000663_0Topic

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

Levelm71c5f897fa3066bd_1528449084556_0Level

Third

Core curriculumm71c5f897fa3066bd_1528449076687_0Core curriculum

I. Real numbers. The student:

1) performs operations (addition, subtraction, multiplication, division, exponentiation, square rooting, logarithms) in the set of real numbers;

9) uses the logarithmic relationship with exponentiation, uses the logarithm of the product, the logarithm of the quotient and the logarithm of power.

Timingm71c5f897fa3066bd_1528449068082_0Timing

45 minutes

General objectivem71c5f897fa3066bd_1528449523725_0General objective

Interpreting and manipulating information presented in the text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables.

Specific objectivesm71c5f897fa3066bd_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics, scientific, technical and IT competences, developing learning skills.

2. Understanding formulas for the logarithm of the product and the logarithm of power.

3. Applying formulas for the logarithm of the product and the logarithm of power.

Learning outcomesm71c5f897fa3066bd_1528450430307_0Learning outcomes

The student:

- learns formulas for the logarithm of the product and the logarithm of power,

- applies formulas for the logarithm of the product and the logarithm of power.

Methodsm71c5f897fa3066bd_1528449534267_0Methods

1. Mind map.

2. Case study.

Forms of workm71c5f897fa3066bd_1528449514617_0Forms of work

1. Individual work.

2. Work in a small group.

Lesson stages

Introductionm71c5f897fa3066bd_1528450127855_0Introduction

The students work in two groups. The first group creates a mind map containing information about logarithms. The second group creates a mind map containing information about operations on powers.

When they finish they present their results.

Procedurem71c5f897fa3066bd_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to learn and use in calculations the formulas for the logarithmlogarithmlogarithm of the product and the logarithm of power.

Students, working in groups, analyze the Slideshow showing the way of obtaining the formula for the logarithm of the product and the logarithm of power. They write the appropriate theorems.

[Slideshow]

Theorem - the logarithm of the productlogarithm of the productlogarithm of the product:
With a positive and different from 1 base a, for any numbers x > 0 and y > 0 a formula occurs:

Theorem - the logarithm of powerlogarithm of the powerlogarithm of power:
With a positive and different from 1 basis a, for any numbers x > 0 and any rational number r there is a formula:

Students use the formulas to solve problems.

Task 1

Calculate the value of the terms:

a) ${\log }_{6}2+{\log }_{6}12+{\log }_{6}9$,

b) $\log 4+\log 25$,

c) ${\log }_{15}9+{\log }_{15}25$,

d) ${\log }_{\frac{1}{8}}2+{\log }_{\frac{1}{8}}4$.

Task 2

Calculate the value of the terms:

a) ${\log }_{3}729$,

b) ${\log }_2\frac{1}{32}$,

c) ${\log }_{6}27+{\log }_{6}8$,

d) $2{\log }_{12}2+2{\log }_{12}6$.

Task 3

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

a) ${\log }_{5}14$,

b) ${\log }_{5}56$,

c) ${\log }_{5}70$,

d) ${\log }_5\mathrm{3,5}$.

Task 4

Calculate x:

a) ${\log }_{5}x=2{\log }_{5}4$,

b) ${\log }_{2}x=-3{\log }_2\frac{{\displaystyle 4}}{3}$,

c) ${\log }_{7}x=2{\log }_2\frac{{\displaystyle 3}}{4}$,

d) ${\log }_{\frac{{\displaystyle 1}}{2}}x=-2{\log }_{\frac{{\displaystyle 1}}{2}}7$.

Task 5

Write in the form of a logarithm:

a) $1+{\log }_{2}5-{\log }_{2}10$,

b) $2\log 3+\log 2+1$,

c) $\frac{1}{3}{\log }_{3}2+2$,

d) ${\log }_{4}6+2$.

Task 6

Prove that for any positive real numbers x and y the given equality is true:m71c5f897fa3066bd_1527752256679_0Prove that for any positive real numbers x and y the given equality is true:

a) $\log \frac{x}{y}+\log \frac{y}{x}=0$,

b) ${\log }_2{x}^{2}y-{\log }_{2}x{y}^2={\log }_{2}x-{\log }_{2}y$.

An extra task

Calculate x:

a) ${\log }_{6}x=2{\log }_{6}4-{\log }_{6}18$,

b) ${\log }_{3}x=-2+2{\log }_{3}2$.

Lesson summarym71c5f897fa3066bd_1528450119332_0Lesson summary

Students do the revision exercises. Together, they formulate theorems to remember.

- With a positive and different from 1 base a, for any number x> 0 and y> 0 the following equality is true.m71c5f897fa3066bd_1527752263647_0- With a positive and different from 1 base a, for any number x> 0 and y> 0 the following equality is true.
${\log }_a(x·y)={\log }_{a}x+{\log }_{a}y$.

- With a positive and different from 1 base a, for any number x > 0 the following equality is true.
${\log }_a{x}^r=r·{\log }_{a}x$.

Selected words and expressions used in the lesson plan

logarithmlogarithm logarithm

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

power of powerpower of powerpower of power

exponentexponentexponent

base of logarithmbase of logarithmbase of logarithm