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## Algebra 2

### Course: Algebra 2>Unit 8

Lesson 1: Introduction to logarithms

# Intro to logarithms

Sal explains what logarithms are and gives a few examples of finding logarithms. Created by Sal Khan.

## Want to join the conversation?

• is there any synonym of logarithm? •   Logarithm is based on the combination of two Greek words: logos and arithmos (number). Logos (λόγος) is a rather curious Greek word with multiple meanings. In this case, you could translate it as "ratio" or "proportion". The word "logarithm" was invented by John Napier in 1614.
• Who invented logarithms? And for what reason? •   Sir John Napier did... It was actually for ease of calculations, when we didn't have digital calculators. Logarithms made it easy for people to carry out otherwise difficult operations, eg: find the value of 4th root of 24. we can simply take log(24) and divide by 4. The antilog of the resultant figure will give us the answer. This is quite a feat, considering that we are not using any calculator!
• Is log base 1 of 1 equal to 1, 0, or both? • In what grade do you learn logarithims? • what is zero to the zero power? • That depends on whom you ask:
There are several very important mathematical theorems that require 0⁰ = 1. Thus, many or most mathematicians have simply defined 0⁰ = 1 without formal proof.

There are a few mathematicians who disagree and assert that 0⁰ is indeterminate or undefined. Particularly in certain constructions that come up in calculus (but not at this level of study) 0⁰ is a form that is indeterminate (for which there are ways to work around it and find a solution).

Thus, it really just depends on whom you ask. Most likely, your instructor would assert 0⁰ = 1, but you should ask her to see, to make sure she is not one of those who dissent from the majority of mathematicians.

NOTE: To date, no mathematician has ever been able to formally prove what, if anything, 0⁰ equals. The assertion that it equals 1 is done so that exceptions do not have to be made to those important theorems.
• What is the purpose of logarithms? • Logarithms serve several important purposes in mathematics, science, engineering, and various fields. Some of their main purposes include:

Solving Exponential Equations: Logarithms provide a way to solve equations involving exponents. When you have an equation of the form a^x = b, taking the logarithm of both sides allows you to solve for x. This is particularly useful when dealing with exponential growth or decay problems.

Simplifying Complex Calculations: Logarithms can simplify computations, especially when dealing with large numbers or complicated mathematical operations. Multiplication and division of numbers can be converted to addition and subtraction, respectively, using logarithmic properties.

Measuring Relative Magnitudes: Logarithms allow us to express large ranges of numbers in a more manageable form. For example, the Richter scale uses logarithms to quantify the energy released by earthquakes, and the pH scale uses logarithms to measure the acidity of a solution.

Decibel Scale: In acoustics and electronics, the decibel scale is used to express the ratio of two quantities, such as sound or power levels. It is based on logarithms and provides a more intuitive representation of relative loudness or signal strength.

Compounding Interest: Logarithms are used in finance to calculate compound interest. They help determine how an investment grows over time, accounting for interest being added to the initial investment and the interest already earned.

Data Compression: Logarithms are employed in data compression algorithms to reduce the size of data for efficient storage and transmission. They enable lossless compression techniques that can later be reconstructed to the original data.

Frequency Analysis: Logarithms are used in signal processing and frequency analysis to convert exponential growth or decay relationships into linear ones. This simplifies the analysis of signals in various applications.

Probability and Statistics: Logarithms are used in probability and statistics to transform skewed data or convert multiplicative relationships into additive ones, making them more amenable to certain statistical techniques.

Machine Learning and Data Science: Logarithms find applications in various machine learning algorithms and data analysis tasks, like feature scaling, transforming skewed distributions, and handling large ranges of numerical data.

Overall, logarithms provide a powerful toolset for dealing with exponential relationships, making computations more manageable, and offering insights into various phenomena across different disciplines. They are an essential mathematical concept with diverse applications in the real world.

But in real life absolutely nothing
• Why x+(1/x) >=2 if x is positive and x+(1/x)<=-2 if x is negative real number?
(1 vote) • First, let's consider the case that x is positive. Note that sqrt(x) is real and nonzero since x is positive.
To show x + (1/x) >=2, it is enough to show that x + (1/x) - 2 >= 0.
The trick is to express x + (1/x) - 2 as a perfect square trinomial!
x + (1/x) - 2 = [sqrt(x)]^2 + [1/sqrt(x)]^2 - 2
= [sqrt(x)]^2 + [1/sqrt(x)]^2 - 2sqrt(x)*[1/sqrt(x)]
= [sqrt(x)]^2 - 2sqrt(x)*[1/sqrt(x)] + [1/sqrt(x)]^2
= [sqrt(x) - 1/sqrt(x)]^2
>=0 since sqrt(x) - 1/sqrt(x) is real (because x is positive).
So x + (1/x) - 2 >= 0, which then implies x + (1/x) >= 2.

Now, let's consider the case that x is negative. Then -x is positive. From the result of the first case using -x in place of x, we have
-x + [1/(-x)] >= 2 which implies -x - (1/x) >= 2 which implies -[x + (1/x)] >= 2 which implies
x + (1/x) <= -2.

Have a blessed, wonderful day!   