Logarithms - The Easy Way!

The Organic Chemistry Tutor
27 May 201710:20
EducationalLearning
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TLDRThis educational lesson focuses on evaluating logarithms with various bases. The instructor explains how to calculate logarithms by determining the power to which a base must be raised to yield a given number. Examples include finding log base 2 of 16, log base 3 of 27, and log base 10 of 1000, among others. The video also covers the properties of logarithms, such as the log of one being zero and the log of a number with two zeros being twice the number of zeros. The lesson helps students understand the concept of logarithms, how to work with negative exponents, and provides practice problems to reinforce learning.

Takeaways
  • ๐Ÿ“š A log is the power to which a base number must be raised to obtain a given number.
  • ๐Ÿ”ข Log base 2 of 16 is 4, because 2^4 = 16.
  • ๐Ÿ“ˆ Log base 3 of 27 is 3, as 3^3 = 27.
  • ๐Ÿ”‘ Log base 5 of 25 is 2, since 5^2 = 25.
  • โšช Log base 4 of 1 is 0 because any non-zero number to the power of 0 is 1.
  • ๐Ÿ”Ÿ Log base 7 of 7 is 1, as 7^1 = 7.
  • ๐Ÿ“‰ Log base 10 of 0.0001 (ten to the negative fourth) is -4.
  • ๐Ÿ”ข Log base 10 of 1000 is 3 because 10^3 = 1000.
  • ๐Ÿ“ˆ Log base 10 of 100 is 2, as 10^2 = 100.
  • ๐Ÿ“š The base of a log is assumed to be 10 if no base is specified.
  • ๐Ÿ“‰ Logarithms of numbers less than 1 are negative, indicating the power to which the base must be raised to get a fraction.
Q & A
  • What is the value of log base 2 of 16?

    -The value of log base 2 of 16 is 4, because 2 multiplied by itself 4 times (2^4) equals 16.

  • How many threes do you need to multiply to get to 27?

    -You need to multiply three 3's to get to 27, as 3 multiplied by itself 3 times (3^3) equals 27.

  • What is the log base 5 of 25?

    -The log base 5 of 25 is 2, because 5 multiplied by itself 2 times (5^2) equals 25.

  • Why is log base 4 of 1 equal to zero?

    -Log base 4 of 1 is equal to zero because any non-zero number to the power of 0 is 1, so 4^0 = 1.

  • What is the value of log base 7 of 7?

    -The value of log base 7 of 7 is 1, because 7 to the first power (7^1) equals 7.

  • If no base is given for a logarithm, what is the default base assumed?

    -If no base is given for a logarithm, the default base assumed is 10.

  • What is the value of log base 10 of 1000?

    -The value of log base 10 of 1000 is 3, because 10 multiplied by itself 3 times (10^3) equals 1000.

  • What is the value of log base 10 of 0.0001?

    -The value of log base 10 of 0.0001 is -4, because 10 to the negative fourth power (10^-4) equals 0.0001.

  • What is the value of log base 7 of 38?

    -The value of log base 7 of 38 is 38, because the base (7) cancels out, leaving the number as the result.

  • What is the value of 5 log base 5 of 14?

    -The value is 14, as the base (5) and the logarithm cancel each other out, leaving the number as the result.

  • What is the value of log base 3 of 9?

    -The value of log base 3 of 9 is 2, because 3 multiplied by itself 2 times (3^2) equals 9.

  • What happens to the exponent when the base and the result of a logarithm are reversed?

    -When the base and the result of a logarithm are reversed, the exponent becomes the reciprocal, and the base and result swap places.

  • What is the value of log base 2 of 32?

    -The value of log base 2 of 32 is 5, because 2 multiplied by itself 5 times (2^5) equals 32.

  • What is the value of log base 32 of 2?

    -The value is 1/5, because 32 to the power of 1/5 equals 2, so the logarithm base 32 of 2 is the reciprocal of 5.

  • How can you determine the sign of the logarithm result when dealing with fractions?

    -The sign of the logarithm result will be negative when the base is greater than the number, resulting in a fraction.

  • What is the value of log base 9 of 1/3?

    -The value is -1/2, because when you reverse the base and the number and take the reciprocal of the exponent, you get 1/3.

Outlines
00:00
๐Ÿ“š Understanding Logarithms and Their Properties

This paragraph introduces the concept of evaluating logarithms with various bases. It begins by explaining how to find the base 2 logarithm of 16, which is determined by multiplying 2 by itself four times to get 16, thus log base 2 of 16 equals four. The paragraph continues with examples using different bases, such as base 3 for 27, base 5 for 25, and base 4 for 1, illustrating the process of finding the exponent by multiplying the base to reach the given number. It also discusses special cases like log base 7 of 7, where the base and the number are the same, resulting in a logarithm of 1. The paragraph further explains the default base of 10 when no base is specified, using examples like log base 10 of 1000 and 100 to demonstrate how many times the base must be multiplied to reach the number. Negative logarithms are also covered, such as log base 10 of 0.0001, which equals negative four. The summary concludes with patterns observed in logarithms, like the number of zeros in the number correlating with the logarithm value, and provides practice problems to reinforce the concepts.

05:01
๐Ÿ” Advanced Logarithm Concepts and Practice Problems

The second paragraph delves deeper into logarithmic concepts, starting with the logarithm of a number with its base, such as 7 log base 7 of 38, which simplifies to 38. It then explores the effects of negative exponents and fractions, like log base 3 of 1/9, which equals negative two, and the relationship between reversing the base and the number, such as log base 9 of 3 resulting in 1/2. The paragraph also discusses the implications of comparing numbers to the base, such as log base 2 of 32 being 5, because 2 multiplied by itself five times is 32, and the reciprocal relationship when dealing with fractions, like log base 32 of 2 being 1/5. The summary includes a variety of practice problems that involve finding powers and roots, such as determining that 2 to the third power is 8, and the square root of 64 is 8, to reinforce the understanding of logarithms and their applications. The paragraph concludes with a set of mixed practice problems to test the viewer's grasp of the concepts presented.

Mindmap
Keywords
๐Ÿ’กLogarithm
A logarithm is the inverse operation to exponentiation, which asks the question 'to what power must a number be raised to obtain a certain value?'. In the context of the video, logarithms are used to solve problems involving powers of numbers. For example, the script explains that 'log base 2 of 16' is asking how many times you must multiply 2 by itself to get 16, which is 4, hence the logarithm is 4.
๐Ÿ’กBase
The base in a logarithmic expression is the number that is being raised to a power. The video emphasizes the importance of the base in determining the result of a logarithm. For instance, 'log base 3 of 27' involves using 3 as the base to find the exponent that results in 27, which is 3.
๐Ÿ’กExponent
An exponent is a number that indicates how many times a base number is multiplied by itself. The video script uses exponents to demonstrate the relationship between logarithms and powers. For example, '2 to the fourth is 16' means that 2 is multiplied by itself four times (2*2*2*2) to get 16.
๐Ÿ’กMultiplication
Multiplication is the operation of adding a number to itself a certain number of times. In the video, multiplication is used to determine the number of times a base must be used to reach a particular value, which is key to understanding logarithms. For example, '2 times 2 times 2 times 2' is the process of multiplying 2 by itself four times to reach 16.
๐Ÿ’กPower
A power is the result of multiplying a number by itself a certain number of times, which is the same as raising a number to an exponent. The video script uses the concept of power to explain logarithms, such as 'three squared is nine', where 'squared' means the base (3) is raised to the power of 2 (3*3).
๐Ÿ’กNegative Exponent
A negative exponent indicates that a number is to be divided into 1 a certain number of times. In the video, negative exponents are used to express the inverse relationship between exponents and logarithms. For example, 'three to the negative two' means 1 divided by 3 squared (1/9).
๐Ÿ’กFraction
A fraction represents a part of a whole, and in the context of logarithms, it can represent the result when a number is raised to a negative exponent. The video explains that when the base and the result are reversed, such as 'log base 9 of 3', the result is a fraction, specifically 1/2.
๐Ÿ’กZero
The script mentions that the logarithm of one is always zero, which is a fundamental property of logarithms. This is because any number raised to the power of zero equals one, so the logarithm base that results in one is zero.
๐Ÿ’กOne Million
The video uses the example of one million to illustrate how logarithms can be used with large numbers. It explains that 'log of one million' equals six because one million has six zeros, and thus, 10 to the power of six equals one million.
๐Ÿ’กNegative Logarithm
A negative logarithm occurs when the result of the logarithm is a negative number. The video script explains that negative logarithms are used to express numbers less than one, such as 'log of .0001', which is '-4' because 10 to the power of -4 equals .0001.
๐Ÿ’กPractice Problems
Practice problems are used in the video to reinforce the understanding of logarithms. They provide viewers with an opportunity to apply the concepts learned in the video to solve various logarithmic expressions, such as '7 log base 7 of 38', which simplifies to 38 by canceling out the base.
Highlights

Definition and calculation of logarithms with base 2, explaining log base 2 of 16 equals four.

How to determine log base 3 of 27, which is 3.

Calculating log base 5 of 25, which equals two since 5 squared is 25.

Understanding log base four of one is always zero.

Explanation of log base seven of seven equals one due to the base and number being the same.

Clarification that the base is assumed to be ten when not specified, and calculation of log base 10 of a thousand.

Calculation of log base 10 of 100, which is two.

Demonstration of how to calculate log base 10 of 0.0001, which is negative four.

Pattern recognition in logarithms: log of a hundred is two, log of a thousand is three.

Introduction of negative logarithms with examples of log of 0.1, 0.01, and 0.001.

Simplification of logarithmic expressions when the base and number are the same, such as 7 log base 7 of 38 equals 38.

How to find 5 log base 5 of 14 and 8 log base 8 of y, which are 14 and y respectively.

Calculation of log base 3 of 9, which is two, since 3 squared is 9.

Understanding the change in logarithm when the base and number are inverted, such as log base 3 of 1/9 equals negative two.

Reversing the base and number in a logarithm and its effect on the result, shown with log base 9 of 3.

Explanation of the logarithm of fractions and how it results in negative values, using log base 9 of 1/3.

General rule for logarithms when the number is larger than the base, resulting in values greater than one.

Examples of logarithms with base 2 and base 32, including calculations for 32, 1/32, and their reciprocals.

Practice problems involving powers and roots of numbers, such as two to the power of three equals eight.

Calculation of negative exponents and their effects, like three to the negative two equals 1/9.

Finding roots and their reciprocals, such as the square root of 64 is eight, and 64 to the power of 1/2 is eight.

Using the fourth root to find the base when the number is a power of that base, shown with 16 and 81.

Transcripts
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