Topicm57e5c7ab6a95712a_1528449000663_0Topic

Definition of logarithm. Properties of logarithmlogarithmlogarithm

Levelm57e5c7ab6a95712a_1528449084556_0Level

Third

Core curriculumm57e5c7ab6a95712a_1528449076687_0Core curriculum

I. Real numbers. Student:

1. performs mathematical operations (addition, subtraction, multiplication, division, exponentiation, root extraction, logarithmlogarithmlogarithm) in a set of real numbers;

9. uses the logarithmlogarithmlogarithmic relationship with exponentiation, uses the formulas for the logarithm of the product, the logarithmlogarithmlogarithm of the quotient and the logarithm of power.

Timingm57e5c7ab6a95712a_1528449068082_0Timing

45 minutes

General objectivem57e5c7ab6a95712a_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 objectivesm57e5c7ab6a95712a_1528449552113_0Specific objectives

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

2. Understanding the concept of logarithmlogarithmlogarithm and its properties.

3. Calculation of logarithm values using logarithmlogarithmlogarithm definitions.

Learning outcomesm57e5c7ab6a95712a_1528450430307_0Learning outcomes

The student:

- applies the concept of logarithmlogarithmlogarithm and its properties,

- calculates log values based on definitions.

Methodsm57e5c7ab6a95712a_1528449534267_0Methods

1. Mind map.

2. Case study.

Forms of workm57e5c7ab6a95712a_1528449514617_0Forms of work

1. Individual.

2. Group work.

Lesson stages

Introductionm57e5c7ab6a95712a_1528450127855_0Introduction

Students, working in groups, organize their knowledge about the exponential functionexponential functionexponential function. They create mind maps. After finishing the task, they present their results.

Procedurem57e5c7ab6a95712a_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to get to know the definition of logarithm and its properties.

Students, working in two groups, use the properties of the exponential functionexponential functionexponential function, to build an intuitive concept of logarithmlogarithmlogarithm.

Task - Group 1
Draw a graph of the exponential function $f\left(x\right)=3^x,x\in R$. Read from the graph the exact values of the arguments for which the function values are equal to $1,3,4,5,9$. Was it always possible? Formulate the conclusion.m57e5c7ab6a95712a_1527752263647_0Draw a graph of the exponential function $f\left(x\right)=3^x,x\in R$. Read from the graph the exact values of the arguments for which the function values are equal to $1,3,4,5,9$. Was it always possible? Formulate the conclusion.

Task - Group 2
Draw a graph of the exponential functionexponential functionexponential function $f\left(x\right)=(\frac{1}{3}{)}^x,x\in R.$ Was it always possible? Formulate the conclusions. Read from the graph the exact values of the arguments for which the function values are equal to $1,3,4,5,9.$ Was it always possible? Formulate the conclusions.

Conclusions

The exact value of the argument could be read only if the function value was a rational power of the number 3 or $\left(\frac{1}{3}\right)$. In other cases, only an approximate value can be given.

The teacher informs students that in cases where it is not possible to accurately read the value of the argument from the graph, the operation  called logarithmlogarithmlogarithm needs to be used.

Students, working independently, analyze SLIDESHOW illustrating the concept of logarithm. They formulate the definition of logarithmlogarithmlogarithm.

Task 1
[Slideshow]

Analyze the material contained in the slideshow very carefully. Formulate the definition of logarithm. Describe the logarithm properties resulting from this definition.m57e5c7ab6a95712a_1527752256679_0Analyze the material contained in the slideshow very carefully. Formulate the definition of logarithm. Describe the logarithm properties resulting from this definition.

Definition

LogarithmlogarithmLogarithm of a positive number c with a positive and different from 1 base a, is the exponent b to which a must be raised to produce c.

We write: ${\log }_{a}c=b$ if and only if $a^b=c$.

The number c is an argument of logarithmargument of logarithmargument of logarithm.

By definition, it appears that:

a. ${\log }_a{a}^c=c$

b. $a^{{\log }_{a}c}=c$

c. ${\log }_{a}1=0$

d. ${\log }_{a}a=1$

Students solve using the algorithm definition.

Task 2
Calculate according to the formula.

1. ${\log }_{2}128$

2. ${\log }_{\sqrt{3}}3\sqrt{3}$

3. ${\log }_{2}64$

4. ${\log }_3\frac{1}{81}$

5. ${\log }_{\frac{1}{3}}\sqrt{3}$

6. ${\log }_{\frac{1}{2}}16$

7. ${\log }_{5}625$

8. ${\log }_6\sqrt{36}$

Task 3
Calculate logarithm base.

1. ${\log }_{a}27=-3$

2. ${\log }_a\frac{1}{64}=3$

3. ${\log }_{a}0,25=-1$

4. ${\log }_a\frac{1}{256}=2$

5. ${\log }_{a}3=\frac{1}{2}$

6. ${\log }_{a}2=\frac{1}{3}$

Task 4
Calculate the value of the expression.

1. ${\log }_{2}8+{\log }_{2}16$

2. ${\log }_{4}4-{\log }_{4}64$

3. ${\log }_{6}6\sqrt{6}-{\log }_{\frac{1}{6}}36$

Task 5
Knowing that $m>2$, calculate ${\log }_m\sqrt[m]{m}$.

An extra task
Specify the domaindomaindomain of the expression ${\log }_{\frac{1}{3}}(2x-4)$.

Lesson summarym57e5c7ab6a95712a_1528450119332_0Lesson summary

Students do revision exercises.

They formulate a definition to remember.

- LogarithmlogarithmLogarithm of a positive number c with a positive and different from 1 base a is the exponent b to which a must be raised to produce c.

Selected words and expressions used in the lesson plan

argument of logarithmargument of logarithmargument of logarithm,

argument of the functionargument of the functionargument of the function,

domaindomaindomain,

exponential functionexponential functionexponential function,

logarithmlogarithmlogarithm,

logarithm baselogarithm baselogarithm base,

value of the functionvalue of the functionvalue of the function