That's Not A Property Of Logarithms!

Algebra Level 1

True or False: $\log(1+2+3) = \log 1 + \log 2 + \log 3$

True False

3 solutions

Michael Fuller
May 8, 2016

Relevant wiki: Properties of Logarithms - Basic

$\log{(1+2+3)}=\log6 \, , \quad \log1 + \log2 + \log 3 = \log{(1 \times 2 \times 3)} = \log6$

Luckily, its true in this case, since $(1,2,3)$ is a solution to the equation $x+y+z=xyz$ :P

Nihar Mahajan - 5 years, 1 month ago

It's the only positive solution. :P

Sharky Kesa - 5 years, 1 month ago

The only positive integer solution. There are infinitely many positive real solutions.

Ivan Koswara - 5 years, 1 month ago

@Ivan Koswara – The only positive integer solution with exactly 3 terms .

Kobe Cheung - 5 years, 1 month ago

@Kobe Cheung – Well, the equation is $x+y+z = xyz$ and not $a_1 + a_2 + \ldots + a_n = a_1 a_2 \ldots a_n$ , so it clearly has 3 terms.

Ivan Koswara - 5 years, 1 month ago

It is true that 1x2x3= 1 + 2 + 3=6. But, log6 > log1 + log2 + log3, because log1=0. The logarithm of the sum of numbers is not always equal to the sum of the individual logarithms of those same numbers. Thus, what you have done here only works for simple arithmetic, not logarithms.

Solomon Hailu - 5 years, 1 month ago

Incorrect; $\log 6 = \log (1 \cdot 2 \cdot 3) = \log 1 + \log 2 + \log 3$ . Remember the property of logarithms $\log ab = \log a + \log b$ .

Ivan Koswara - 5 years, 1 month ago

i was thinking this^

Bryan Gonzalez - 5 years, 1 month ago

Akash Patalwanshi
May 8, 2016

$\large \log(1+2+3) = \log6 = \log(1*2*3)$

$\large = \log1 + \log2 + \log3$

When does inner addition equal separated addition in logs? I thought the rule was inner multiplication = separated addition and inner division equals separated subtraction? :o

Bryan Gonzalez - 5 years, 1 month ago

Inner addition equals to separated addition of logs in this case because $1 + 2 + 3 = 1 * 2 * 3$

akash patalwanshi - 5 years, 1 month ago

Shreshth Goyal
May 29, 2016

Given, Log (1+2+3)= log 6 Log 6 can also be written as log 2 × 3 Log 2 × 3 = log 2 + log 3

We know that for any 'log 1' is always equal to '0'...therefore if we add log 1 it won't make much difference.

So log (1+2+3) = log 1 + log 2 + log 3 ☝ this is not a property of logarithms. But in this question it is equal to 'log 1.......log 3'

That's Not A Property Of Logarithms!

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