Log Number

Algebra Level 2

Suppose a real number $A$ satisfy the inequality $8^{21} < A < 8^{22}$ and $\log _{ 2 }{ A }$ is an even number, find the value of $\log _{ 4 }{ A }$ .

Image Credit: Flickr Dominic Alves .

The answer is 32.

1 solution

Ravi Dwivedi
Jul 11, 2015

$8^{21}<A<8^{22}\\ \implies 2^{63} < A < 2^{66}\\$

Taking log of base 2 both sides we get

$63<\log_2 A<66\\$

(Note: base>1 so inequality remains same)

Since $log_2 A$ is even and the only even number between $63$ and $66$ is $64$ it follows that

$log_2 A = 64\\$

$A=2^{64}=4^{32}\\$

$log_4 A = \boxed{32}$

Moderator note:

To be precise, you should rephrase it to "The only even number in between 63 and 66 is 64."

Log Number

Image Credit: Flickr Dominic Alves .

The answer is 32.

1 solution

Moderator note:

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