Logarithms Are Tricky

Algebra Level 2

5 5 = 3125 ⇒ log ⁡ 5 3125 = 5 4 5 = 1024 ⇒ log ⁡ 4 1024 = 5 3 5 = 243 ⇒ log ⁡ 3 243 = 5 2 5 = 32 ⇒ log ⁡ 2 32 = 5 \large{ \begin{array}{lll} &5^5 = 3125 &\Rightarrow &\log_5 3125 = 5 \\ &4^5 = 1024 &\Rightarrow &\log_4 1024 = 5 \\ &3^5 = 243 &\Rightarrow &\log_3 243 = 5 \\ &2^5 = 32 &\Rightarrow &\log_2 32 = 5 \end{array}}

The above mathematical working are all correct.
Given that 1 5 = 1 1^5 = 1 , is it true that log ⁡ 1 1 = 5 \log_1 1 = 5 ?

True False

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Chung Kevin by Chung Kevin , 45, Australia

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9 solutions

log ⁡ 1 \log_1 does not exist.

7 Helpful 0 Interesting 0 Brilliant 0 Confused
Johnny Jillky
Mar 19, 2016

Further, would log ⁡ 1 1 \log_1 1 be indeterminate?

2 Helpful 0 Interesting 0 Brilliant 0 Confused

More like undefined.

Whitney Clark - 5 years, 2 months ago

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Whoops mixed up my terminology there

Johnny Jillky - 5 years, 2 months ago
Kay Xspre
Mar 18, 2016

No. Logarithm is general sense is defined for positive real not being in the base of 1.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Take note log ⁡ 1 1 = ∞ \log_{1}1 = \infty

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Pshanthi Radha
Mar 28, 2016

Log 1 to any base is zero.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Except for base 1 as shown in this problem.

Andy Wong - 5 years, 2 months ago
Ska Bz
Mar 28, 2016

log_b(a) = ln(a)/ln(b) ln(b) different to 0 => b different to 1 !

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Sagar Kothari
Mar 28, 2016

The only possible log function that yield one would be if the base is 0.

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Kim Hale
Mar 28, 2016

y=0 always

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Harshendu Mahto
Mar 19, 2016

For any number of the form :

log ⁡ x 1 = y \log_x 1 = y

y = 0....... in all cases

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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