Question

Write $2\log 3+3\log 5+5\log 2$ as a single logarithm.

Answer 1

Write $2\log 3+3\log 5+5\log 2$ as a single logarithm.

Use, $n\log m=\log m^n$

Then,

$2\log 3+3\log 5+5\log 2=\log \left(3^2\right)+\log \left(5^2\right)+\log \left(2^5\right)$

Use, $\log a+\log b+\log c=\log \left(abc\right)$

Then,

$\begin{array}{rcl}2\log 3+3\log 5+5\log 2& =& \log \left(3^2\right)+\log \left(5^2\right)+\log \left(2^5\right)\\ & =& \log \left(3^2\times 5^2\times 2^5\right)\\ & =& \log \left(9\times 25\times 32\right)\\ & =& \log \left(9\times 25\times 32\right)\\ & =& \log \left(3600\right)\end{array}$

Thus, $2\log 3+3\log 5+5\log 2$ can be written as a single logarithm as $\log \left(36000\right)$.