We will use the following formulas to solve the problems of logarithms.
• Product Rule of Logarithm: $\log_a(MN)=\log_a M +\log_a N$
• Quotient Rule of Logarithm: $\log_a(M/N)=\log_a M -\log_a N$
• Power Rule of Logarithm: $\log_a M^k=k\log_a M$
• Base Change Rule of Logarithm: $\log_a M=\log_b M \cdot \log_a b$
Problem 1: Find the value of $\log_3 \log_3 27$
Solution: Note that $27=3^3$
∴ $\log_3 \log_3 27$ $=\log_3 \log_3 3^3$ $=\log_3 3$ $[\because \log_a a^n=n]$
$=1$ $[\because \log_a a=1]$
So $\log_3 \log_3 27=1$
…to be continued.
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