MHB Higher Prelim Exam: Solving Log Questions

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To solve the Higher Prelim exam log questions, the definition of a logarithm is crucial, where log_a(b) = c indicates that a^c = b. For part (a), given log_4(x) = P, it can be shown that log_16(x) equals 1/2P by recognizing that log_16 is the logarithm with base 4 squared. In part (b), the equation log_3(x) + log_9(x) = 12 can be simplified using the properties of logarithms, specifically that log_9(x) can be expressed in terms of log_3(x). Understanding these logarithmic properties is essential for solving the problems effectively.
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A Question from Em on Yahoo answers:

Please can someone help with this question from a Higher Prelim paper? [A knowledge of how to change log bases is not a requirement of the syllabus.]7(a) Given that log_4(x)=P, show that log_16(x) =1/2P
(b) Solve log_3(x)+log_9(x)=12
 
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CaptainBlack said:
A Question from Em on Yahoo answers:

Please can someone help with this question from a Higher Prelim paper? [A knowledge of how to change log bases is not a requirement of the syllabus.]7(a) Given that log_4(x)=P, show that log_16(x) =1/2P
(b) Solve log_3(x)+log_9(x)=12
What you need to use is the definition of a logarithm: \( \log_a (b)=c \) means that \( a^c=b \).

The other thing you need to do these is to observe that:
\[ (a^2)^{c/2}=a^c=b \]
so:
\[\log_{a^2} (b)=c/2 =\frac{ \log_a (b)}{2}\]
Which is as far as I will go as this is an exam questionCB
 
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