Question

Write 3 log5+4 log6-5 log3 as a single logarithum.​

Answer 1

$\textsf{\large{\underline{Solution}:}}$

We have to write the given expression as a single logarithm.

We know that:

$\rm: \longmapsto \log \bigg( \dfrac{x}{y} \bigg) = \log(x) - \log(y)$

$\rm: \longmapsto \log(xy) = \log(x) + \log(y)$

$\rm: \longmapsto n\log(x) = \log( {x}^{n})$

Therefore, we have:

$\rm = 3 \log(5) + 4 \log(6) - 5 \log(3)$

$\rm = \log( {5}^{3} ) +\log( {6}^{4} ) -\log( {3}^{5} )$

$\rm = \log \bigg( \dfrac{ {5}^{3} \times {6}^{4} }{ {3}^{5} } \bigg)$

$\rm = \log \bigg( \dfrac{ {5}^{3} \times {3}^{4} \times {2}^{4} }{ {3}^{5} } \bigg)$

$\rm = \log \bigg( \dfrac{ {5}^{3} \times {2}^{4} }{ 3 } \bigg)$

$\rm = \log \bigg( \dfrac{ {5}^{3} \times {2}^{3} \times 2 }{ 3 } \bigg)$

$\rm = \log \bigg( \dfrac{{10}^{3} \times 2 }{ 3 } \bigg)$

$\rm = \log \bigg( \dfrac{2000}{ 3 } \bigg)$

★ Which is our required answer.

$\textsf{\large{\underline{More To Know}:}}$

$\rm 1. \: \: {a}^{n} = b \implies log_{a}(b) = n$

$\rm 2. \: \: log_{a}(1) = 0, \: a \neq0,1$

$\rm 3. \: \: log_{a}(a) = 1, \: a \neq0,1$

$\rm 4. \: \: log_{a}(x) = log_{a}(y) \implies x = y$

$\rm 5. \: \: log_{e}(x) = ln(x)$

$\rm6. \: \: log_{a}(x) + log_{a}(y) = log_{a}(xy)$

$\rm7. \: \: log_{a}(x) - log_{a}(y) = log_{a} \bigg( \dfrac{x}{y} \bigg)$

$\rm 8. \: \: log_{a}( {x}^{n} ) = n\log_{a}(x)$

$\rm 9. \: \: log_{a}(m) = \dfrac{ log_{b}(m) }{ log_{b}(a) },m > 0,b > 0,a \ne1,b \ne1$

$\rm 10. \: \: log_{a}(b) = \dfrac{1}{ log_{b}(a) }$