We write $2^3 = 8$ and express the same as $\log (8)$ to the base $2$ is $3$. Isn't this the same as finding nth root of a number? What is the significance of logarithms actually and what if we don't have the concept of Logarithms?
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1Its slightly different - if you start with the number 8, you can either take the logarithm base 2 to get 3, which tells you that $2^3=8$, or you take the nth root where $n=3$ to get 2, which tells you the same thing. So the first way, you start with the numbers 8 and 2 and arrive at 3, while the second way, you start with the numbers 8 and 3 and arrive at 2. They both ultimately give you the same relationship, but you started off with different information to get there. – John Doe Jan 13 '21 at 07:20
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No. The $n$th is knowing the final amount and the power and finding the base. The logartihm is knowing the final amount and the base and finding the power. If you example $\log_2 8 = ??????$ means figuring out $2^{???????} = 8$. But $\sqrt[3]8 = !!!!!!!!$ means figuring out $!!!!!!!^3 = 8$. – fleablood Jan 13 '21 at 07:24
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" Isn't this the same as finding nth root of a number?" well, if you don't know what $n$ is that won't help you. $\log_2 8 = 3$ and $\sqrt[3] 8 =2$ and $2^3 =8$ are indeed equivalent statements and mean the same thing but they answer three different question. $\log_2 8 = 3$ answers "$2$ to what power equals $8$" and $\sqrt[3] 8= 2$ answers "what to the $3$rd power equals $8$". And $2^3 = 8$ answers "what is $2$ to the $3$rd power". Three different questions with three different answers all about the same relation. – fleablood Jan 13 '21 at 07:29
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The b-th root is the answer to Which number should we raise to the power $b$ to get $c$ ? ($a = \sqrt[b]{c}$), while the logarithm is the answer to To which power should we raise the number $a$ to get $c$ ? ($b=\log_a c$).
Gribouillis
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$2^3 = 8$ and $\sqrt[3] 8 = 2$ and $\log_2 8 = 3$ all express the same relationship between $2,3$ and $8$. But the each refer to different aspect of the relation ship.
$\sqrt[3] 8 = \color{red} x$ means to solve $\color{red}x^3 = 8$. We know what the final result is, and we know the power, but we don't know the base. That is what $\sqrt[3] 8$ is supposed to express; the base.
$\log_2 8 = \color{green} y$ means to solve $2^{\color{green} y} = 8$. We know what the final result is and we know the base, but we don't know the power. That is what $\log_2 8$ is supposed to express; the power.
And $2^3 = \color{blue} z$ means to solve $2^3 = \color{blue}z$. We know the base and we know the power, but we don't know what the final result. That is what $2^3$ is supposed to express; the final result.
So $2^3 = 8, \sqrt[3]8 = 2; \log_2 8 = 3$. One relationship with three different components, and an expression to describe each possible component.
fleablood
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One historical significance of logarithms is that they change multiplication into addition, and that was used in practical computation (and inherent in the slide rule as a calculating device) before electronic calculators.
Here are some more mathematically advanced aspects of the logarithm, which give a clue as to why the function emerges so often and across a range of mathematical applications:
Logarithms solve the problem of integrating $\frac 1x=x^{-1}$. Other integer powers of $x$ can be integrated by taking the inverse of obvious derivatives.
The natural logarithm is the inverse function of the exponential function ($e^x$), which is a fixed point of the operation of differentiation.
The discrete logarithm (and associated primitive root of a finite field) are significant in coding theory. This replicates the multiplication to addition aspect.
Mark Bennet
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Since no one answered the question in the manner I would've liked to see myself, here's my answer:
base ^ exponent = result/number
where, the expression (base ^ exponent) is called 'power', not the result/number. If you are wondering why the result is not called 'power', check this link.
For example:
2 ^ 3 = 8
where, 2 is the 'base', 3 is the 'exponent', the expression (2^3) is the 'power' and 8 is the result/number.
With these terminologies in mind -
Finding log of a number, given a base, is to find the exponent => log base 2 of 8 = 3.
Finding nth root or bth root or < whatever >th root of a number, given an exponent, is to find the base => 3rd root of 8 = cube root of 8 = 2.
Hope this helps.
ZeZNiQ
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