Properties of logarithms: Difference between revisions
(Created page with "We define <math>b^n = a</math> as number <math>b</math> to the power <math>n</math> is equal to <math>a</math>. With zero to the power zero being undefined. The logarithm is defined as <math>\log_{b}a = n</math>. That is, the number <math>a</math>, with a base <math>b</math>, such that <math>b^n = a</math>. With exponentiation we want to find the result of the operation. With logarithm we want to find the exponent itself by knowing the base and the result of the exponent...") |
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We define <math>b^n = a</math> as number <math>b</math> to the power <math>n</math> is equal to <math>a</math>. With zero to the power zero being undefined. The logarithm is defined as <math>\log_{b}a = n</math>. That is, the number <math>a</math>, with a base <math>b</math>, such that <math>b^n = a</math>. With exponentiation we want to find the result of the operation. With logarithm we want to find the exponent itself by knowing the base and the result of the exponentiation. | We define <math>b^n = a</math> as number <math>b</math> to the power <math>n</math> is equal to <math>a</math>. With zero to the power zero being undefined. The logarithm is defined as <math>\log_{b}a = n</math>. That is, the number <math>a</math>, with a base <math>b</math>, such that <math>b^n = a</math>. With exponentiation we want to find the result of the operation. With logarithm we want to find the exponent itself by knowing the base and the result of the exponentiation. | ||
I'm going to rely on the arrow and on the function's concept to explain a little confusion that happens with log and exp. Suppose I write this <math>b^n \to a</math> to mean a function that associates two numbers to another. Then I write this <math>a \to b^n</math> to make the reversed association. First, | I'm going to rely on the arrow and on the function's concept to explain a little confusion that happens with log and exp. Suppose I write this <math>b^n \to a</math> to mean a function that associates two numbers to another. Then I write this <math>a \to b^n</math> to make the reversed association. First, <math>b^n</math> has a fixed base, the variable is the power (exponent). Second, log is the inverse of exp, but the reversed arrow exposes a little confusion. What the log tries to find is the unknown power, not the base or <math>a</math> because these two are already known. That's a very common confusion, to think that log is calculating the base, which we already know. We can have unknowns anywhere in the log, be it the base, the exponent or the result. But the way we define log is as the inverse of exponentiation and it only makes sense for this definition is to have the unknown to be the exponent. | ||
'''If you remember that what the log calculates is the exponent from the exponential, that should make all properties of logarithms easier to grasp:''' | |||
==The properties== | |||
Revision as of 16:50, 5 February 2022
We define [math]\displaystyle{ b^n = a }[/math] as number [math]\displaystyle{ b }[/math] to the power [math]\displaystyle{ n }[/math] is equal to [math]\displaystyle{ a }[/math]. With zero to the power zero being undefined. The logarithm is defined as [math]\displaystyle{ \log_{b}a = n }[/math]. That is, the number [math]\displaystyle{ a }[/math], with a base [math]\displaystyle{ b }[/math], such that [math]\displaystyle{ b^n = a }[/math]. With exponentiation we want to find the result of the operation. With logarithm we want to find the exponent itself by knowing the base and the result of the exponentiation.
I'm going to rely on the arrow and on the function's concept to explain a little confusion that happens with log and exp. Suppose I write this [math]\displaystyle{ b^n \to a }[/math] to mean a function that associates two numbers to another. Then I write this [math]\displaystyle{ a \to b^n }[/math] to make the reversed association. First, [math]\displaystyle{ b^n }[/math] has a fixed base, the variable is the power (exponent). Second, log is the inverse of exp, but the reversed arrow exposes a little confusion. What the log tries to find is the unknown power, not the base or [math]\displaystyle{ a }[/math] because these two are already known. That's a very common confusion, to think that log is calculating the base, which we already know. We can have unknowns anywhere in the log, be it the base, the exponent or the result. But the way we define log is as the inverse of exponentiation and it only makes sense for this definition is to have the unknown to be the exponent.
If you remember that what the log calculates is the exponent from the exponential, that should make all properties of logarithms easier to grasp:
