Determine value of the expression 1/logbase9(15) + 1/logbase25(15)
Use the change of base rule: $\displaystyle log_c(a) = \frac{log_b(a)}{log_b(c)}$
$\displaystyle log_{9}(15) = \frac{ln(15)}{ln(9)}$
$\displaystyle log_{25}(15) = \frac{ln(15)}{ln(25)}$
Flipping the two results above and adding gives: $\displaystyle \frac{ln(25)+ln(9)}{ln(15)}$
Simplify using the log multiplication rule: $\displaystyle a\,ln(k) = ln(k^a)$
$\displaystyle \frac{ln(25)+ln(9)}{ln(15)} = \frac{2[ln(5)+ln(3)]}{ln(15)}$
Simplify using the log addition rules: $\displaystyle ln(a)+ln(b) = ln(ab)
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$\displaystyle \frac{2[ln(5)+ln(3)]}{ln(15)} = \frac{2ln(15)}{ln(15)} = 2$