Question

Compare log3 with base 2 with log5 with base 3.

Remember if the question if done by directly substituting the values, it will be consider as spam.

Answer 1

Answer:

log(2 base)3 = { (log3)²÷ log7 } × log(3 base)5

Step-by-step explanation:

Let, log(2 base)3 = n × log(3 base)5

or, (log3 ÷ log2) = n× ( log5÷ log3)

or, n = (log3)² ÷ (log5×log2)

or, n= (log3)² ÷ log7

Answer 2

$\large\underline{\sf{Solution-}}$

Let assume that,

$\rm \: log_{3}(5) \: < \: log_{2}(3)$

can be further rewritten as

$\rm \:iff \: \: 5 < {3}^{ log_{2}(3) }$

On squaring both sides, we get

$\rm \:iff \: \: {5}^{2} < {3}^{ 2 \: log_{2}(3) }$

We know,

$\boxed{\sf{  \:\rm \: y \: log_{a}(x) \: = \: log_{a}( {x}^{y} ) \: \: }}$

So, using this result, we get

$\rm \:iff \: \: 25 < {3}^{ log_{2}( {3}^{2} ) }$

$\rm \: iff \: \: 25 < {3}^{ log_{2}9}$

$\rm \:iff \: \: 25 < {3}^{ log_{2}9} < {3}^{ log_{2}(8) }$

$\rm \: iff \: \: 25 < {3}^{ log_{2}(8) }$

$\rm \: iff \: \: 25 < {3}^{ log_{2}( {2}^{3} ) }$

$\rm \:iff \: \: 25 < {3}^{ 3log_{2}( 2) }$

We know,

$\boxed{\sf{  \:\rm \: log_{x}(x) \: = \: 1 \: \: }}$

So, using this identity, we get

$\rm \:iff \: \: 25 < {3}^{3}$

$\rm \: iff \: \: 25 < 27$

$\rm \: which \: is \: true$

Hence, our assumption is true.

$\rm\implies \:\rm \: log_{3}(5) \: < \: log_{2}(3)$

$\rule{190pt}{2pt}$

Additional Information :-

$\boxed{\sf{  \:\rm \: logx + logy = logxy \: \: }}$

$\boxed{\sf{  \:\rm \: logx - logy = log \frac{x}{y} \: \: }}$

$\boxed{\sf{  \:\rm \: log_{x}(y) = \frac{logy}{logx} \: \: }}$

$\boxed{\sf{  \:\rm \: log_{x}(x) = 1 \: \: }}$

$\boxed{\sf{  \:\rm \: log_{x}( {x}^{y} ) = y \: \: }}$

$\boxed{\sf{  \:\rm \: log_{ {x}^{z} }( {x}^{y} ) = \frac{y}{z} \: \: }}$

$\boxed{\sf{  \:\rm \: log_{ {w}^{z} }( {x}^{y} ) = \frac{y}{z} log_{w}(x) \: \: }}$

$\boxed{\sf{  \:\rm \: {a}^{ log_{a}(x) } \: = \: x \: \: }}$

$\boxed{\sf{  \:\rm \: {a}^{ y \: log_{a}(x) } \: = \: {x}^{y} \: \: }}$