Question

Solve the following
$log_{3 \sqrt{2} }324$
​

Answer 1

Step-by-step explanation:





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>>Fundamental laws of Logarithms

>>Solve log3√(2)5832 | Maths Questions

Question



Solve log325832

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Solution



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3∣5832

_____________

3∣1944

_____________

3∣648

_____________

3∣216

_____________

3∣72

_____________

3∣24

_____________

      8

______________

⇒5832=36×23=(2)6×36=(32)6

⇒log325832=6.

Answer 2

Solution:

Let us assume that:

$\rm \longrightarrow log_{3 \sqrt{2} }(324) = x$

Then:

$\rm \longrightarrow {(3 \sqrt{2} )}^{x} = 324$

Can be written as:

$\rm \longrightarrow {( \sqrt{9} \sqrt{2} )}^{x} = 324$

$\rm \longrightarrow {(\sqrt{18} )}^{x} = 324$

$\rm \longrightarrow {(\sqrt{18} )}^{x} =18 \times 18$

$\rm \longrightarrow {(\sqrt{18} )}^{x} = {18}^{2}$

$\rm \longrightarrow {(18)}^{ \dfrac{x}{2} } = {18}^{2}$

Comparing base, we get:

$\rm \longrightarrow \dfrac{x}{2} = 2$

$\rm \longrightarrow x = 4$

Therefore:

$\rm \longrightarrow log_{3 \sqrt{2} }(324) = 4$

★ Which is our required answer.

Learn More:

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

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

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

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

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

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

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

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

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

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