Properties of logarithms

Algebra Level 5

Let a , b a,b be positive real numbers. Which of the following is not always true?

( A ) : a ln ⁡ ( b ) = b ln ⁡ ( a ) (A) : a^{\ln{(b)}} = b^{\ln{(a)}}

( B ) : ln ⁡ ( a ) = lim ⁡ h → 0 a h − 1 e h − 1 (B) : \ln{(a)} = \lim_{h \to 0} \dfrac{a^h - 1}{e^h - 1}

( C ) : log ⁡ b ( a ) = ln ⁡ ( a ) ln ⁡ ( b ) (C) : \log_b(a) = \dfrac{\ln(a)}{\ln(b)}

( D ) : ln ⁡ ( a b ) = b × ln ⁡ ( a ) (D) : \ln{(a^b)} = b\times \ln(a)

D A B C

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Ariel Gershon by Ariel Gershon , 26, Canada

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1 solution

As log ⁡ 1 ( a ) \log_{1}(a) has no meaning and ln ⁡ ( 1 ) = 0 \ln(1) = 0 , statement (C) would require the additional condition that b ≠ 1 b \ne 1 to be valid. None of the other statements have this "problem" to deal with.

5 Helpful 0 Interesting 0 Brilliant 0 Confused

For option B , if a = 0 then it would also be incorrect

U Z - 6 years, 4 months ago

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That's why it says a , b a,b are positive real numbers

Ariel Gershon - 6 years, 4 months ago

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Oh sorry , did'nt read the question completely , anyway nice question

U Z - 6 years, 4 months ago

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