Question 1202070
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Answer: <font color=red size=4>n = -3</font>


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Work Shown:


0.25 = 1/4


{{{log(b,(x)) = (log(10,(x)))/(log(10,(b)))}}} Change of base formula


{{{log(0.25,(64)) = (log(10,(64)))/(log(10,(0.25)))}}}


{{{log(0.25,(64)) = (log(10,(64)))/(log(10,(1/4)))}}}


{{{log(0.25,(64)) = (log(10,(2^6)))/(log(10,(2^(-2))))}}}


{{{log(0.25,(64)) = (6*log(10,(2)))/(-2*log(10,(2)))}}} Use the log rule log(A^B) = B*log(A)


{{{log(0.25,(64)) = (6*highlight(log(10,(2))))/(-2*highlight(log(10,(2))))}}}


{{{log(0.25,(64)) = (6*cross(log(10,(2))))/(-2*cross(log(10,(2))))}}}


{{{log(0.25,(64)) = (6)/(-2)}}}


{{{log(0.25,(64)) = -3}}}



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Another approach:


{{{log(b,(x)) = (log(10,(x)))/(log(10,(b)))}}} Change of base formula


{{{log(0.25,(64)) = (log(10,(64)))/(log(10,(0.25)))}}}


{{{log(0.25,(64)) = (log(10,(64)))/(log(10,(1/4)))}}}


{{{log(0.25,(64)) = (log(10,(4^3)))/(log(10,(4^(-1))))}}} 


{{{log(0.25,(64)) = (3*log(10,(4)))/(-1*log(10,(4)))}}}


{{{log(0.25,(64)) = (3*highlight(log(10,(4))))/(-1*highlight(log(10,(4))))}}}


{{{log(0.25,(64)) = (3*cross(log(10,(4))))/(-1*cross(log(10,(4))))}}}


{{{log(0.25,(64)) = (3)/(-1)}}}


{{{log(0.25,(64)) = -3}}}


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Another approach:


{{{log(0.25,(64)) = n}}}


{{{64 = 0.25^n}}} Convert to exponential form


{{{64 = (1/4)^n}}}


{{{4^3 = (1/4)^n}}}


{{{4^3 = (4^(-1))^n}}}


{{{4^3 = 4^(-n)}}}


The bases are both 4, so the exponents must be equal.
{{{3 = -n}}}


{{{n = -3}}}
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