Concepto de logaritmo

Summary

This video explains the fundamentals of logarithms, focusing on their definition, base conditions, and how to calculate them using various examples. It covers the relationship between exponents and logarithms, illustrating how to solve for unknowns by comparing powers of numbers. The examples range from simple cases like log base 2 of 8 to more complex ones involving decimals and approximations. The script also introduces the common logarithm (base 10) and explores log base 7 and base 5 problems. Overall, it provides a comprehensive understanding of logarithms and their properties for learners.

Takeaways

• 😀 Logarithms are defined with a base 'a' and an exponent 'r' such that a^r = X.
• 😀 The base of a logarithm must be positive and not equal to 1 for the equation to hold true.
• 😀 In logarithmic notation, the base raised to the exponent equals the number being logged (a^r = X).
• 😀 Example 1: For log base 2 of 8, since 8 = 2^3, the result is 3.
• 😀 Example 2: For log base 5 of 625, since 625 = 5^4, the result is 4.
• 😀 When no base is specified, the logarithm is assumed to be base 10, known as the common logarithm.
• 😀 Example 3: For log base 10 of 100, since 100 = 10^2, the result is 2.
• 😀 If the base and the number are the same, like log base 7 of 7, the result is always 1.
• 😀 Logarithms can work with fractional numbers, like log base 2 of 0.25, where the result is -2.
• 😀 Logarithms can also be approximated when the base and number don't have an exact power relationship, like log base 3 of 11, which is approximately 2.1.
• 😀 The general principle of logarithms is to find the exponent that makes the base raised to that power equal to the number in question.

Q & A

What is the definition of a logarithm as described in the script?

A logarithm has a base, which is denoted as 'a', and it is applied to a number 'X'. The result, 'r', satisfies the equation where the base 'a' raised to the power 'r' equals the number 'X'. Additionally, the base must be a positive number greater than zero and not equal to one.

What are the conditions for the base of a logarithm?

The base of a logarithm must be a positive number greater than zero and cannot be equal to one.

How can we calculate the logarithm of 8 with base 2?

We express 8 as 2^3, so by comparison, the logarithm of 8 with base 2 is 3.

What does it mean when a logarithm does not have an indicated base?

If a logarithm does not have an indicated base, it is assumed to be base 10 by convention, similar to how square roots are understood to be square roots without the symbol for square.

How is the logarithm base 5 of 625 calculated?

To calculate the logarithm base 5 of 625, we factor 625 by repeatedly dividing by 5, getting 625 = 5^4. Therefore, the logarithm base 5 of 625 is 4.

What is the logarithm base 10 of 100?

The logarithm base 10 of 100 is 2, because 100 is equal to 10^2.

What is the result of the logarithm base 7 of 7?

The logarithm base 7 of 7 is 1, because 7 raised to the power of 1 equals 7.

How is the logarithm of 0.25 with an unknown base determined?

For the logarithm of 0.25, we consider the powers of 2. Since 2 raised to the power of -2 equals 0.25, the logarithm of 0.25 with base 2 is -2.

How can the logarithm base 3 of 11 be estimated?

The logarithm base 3 of 11 lies between the logarithms of 3^2 (which is 9) and 3^3 (which is 27). Since 11 is closer to 9, the logarithm is approximately 2.1.

What is the general rule when the base and the number in a logarithmic equation are equal?

When the base and the number in a logarithmic equation are equal, the logarithm equals 1. This is because the base raised to the power of 1 equals the number itself.