Question

Find the logarithm of 64 to the base 2√2 is

Answer 1

Step-by-step explanation:

log

2

2

64=4

Step-by-step explanation:

Let \: x = log_{2\sqrt{2}}64Letx=log

2

2

64

\implies x = log_{2\times 2^{\frac{1}{2}}} 2^{6}⟹x=log

2×2

2

1

2

6

\implies x = log_{2^{\frac{(2+1)}{2}}}2^{6}⟹x=log

2

2

(2+1)

2

6

\implies x = log_{2^{\frac{(3)}{2}}} 2^{6}⟹x=log

2

2

(3)

2

6

\implies x = \frac{6}{\frac{3}{2}}\times log_{2}2⟹x=

2

3

6

×log

2

2

\* we know the logarithmic law:

\boxed {log_{a^{n}}a^{m} = \frac{m}{n}}

log

a

n

a

m

=

n

m

\implies x = 6 \times \frac{2}{3}⟹x=6×

3

2

\implies x = 4⟹x=4

Answer 2

Step-by-step explanation:

$log_{2 \sqrt{2} }(64) \\ = log_{2 \sqrt{2} }( {2 \sqrt{2} } )^{4} \\ = 4 log_{2 \sqrt{2} }(2 \sqrt{2} ) \\ = 4 \frac{ log(2 \sqrt{2} ) }{ log(2 \sqrt{2} ) } \\ = 4$

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