Question

express in terms of log 2 and log3 (i)Log 36 (ii)log144 (iii)Log 4.5 (iv)log 26÷51_log91÷119 (v)log75÷16_2log5÷9+log32÷243

Answer 1

hello users......

Using formula
log mn =log m +log n
& 
log m/n = log m - log n
&
log( m to the power n) = n log m 

Solution :-
(1). log 36 = log 9*4 = log 9+ log 4= log 3² +log2² = 2log 3 +2log 2 answer 

(2). log 144 = log 16*9 = log 16 +log 9 = log 2²×2² +log 3²
 = log 2² +log 2² +log 3² =2log 2 +2 log 2 + 2log3 = 4log 2 + 2log 3 answer

(3). log 4.5 = log 45/10 = log 45 - log 10 = log 5×3² - 1 (this can not be expressed in log 2 ,log 3 )  = log 5 +2log 3 -1 answer 

(4). log 26/51 - log 91/ 119 = log 26/51 ÷ 91/119 = log 26*119 /51*91 =
log 3094 / 4641 = log  2/3 =log 2 -log 3 answer

(5).log 75/16 - 2 log 5/9 + log32/ 243= log(75/16)×(32/243) - log (5/9)²
=  log 150/243 - log 25/49 = log 150/243 ÷ 25/49 = log 150×49/25×243
 =log 6*49/243 = log 2*49 /81 = log 2 +2log 7 -4 log3(this one also can not expressed in log 2 ,log 3) answer

hope it helps ....

Answer 2

Given:

Log 36

log 144

log 4.5

$log \frac {26}{51} - log \frac {91}{119}$

$log \frac {75}{16} - 2 \times log \frac {5}{9} + log \frac {32}{ 243}$

To express:

In terms of log 2 and log 3

Solution:

(i) log 36

36 can be separated as $9 \times 4$

$= log 9 \times 4$

= log 9 + log 4

$= log 3^{2} +log 2^{2}$

$= 2 \times log 3 +2 \times log 2$

(ii) log 144

$= log 16 \times 9$

= log 16 + log 9

$= log 2^{2} \times 2^{2} +log 3^{2}$

$= log 2^{2} + log 2^{2} +log 3^{2}$

$=2 \times log 2 +2 \times log 2 + 2 \times log 3$

$= 4 \times log 2 + 2 \times log 3$

(iii) log 4.5

$= log \frac {45}{10}$

= log 45 - log 10

$= log 5 \times 3^{2} - 1$ (this can’t be expressed in log 2, log 3 )  

$= log 5 +2 \times log 3 -1$

(iv) $log \frac {26}{51} - log \frac {91}{ 119}$

$= log \frac {\frac {26}{51}} {\frac {91}{119}}$

$= log \frac {26 \times 119}{51 \times 91}$

$= log \frac {3094}{4641}$

$= log \frac {2}{3}$

$= log 2 -log 3$

(v) $log \frac {75}{16} - 2 \times log \frac {5}{9} + log \frac {32}{ 243 }$

$= log \frac {75}{16} \times \frac {32}{243} - log {\frac {5}{9}}^{2}$

$= log \frac {150}{243} - log \frac {25}{49}$

$= log \frac {\frac {150}{243}}{\frac {25}{49}}$

$= log \frac {150 \times 49}{25 \times 243}$

$= log 6 \times \frac {49}{243}$

$= log 2 \times \frac {49}{81}$

$= log 2 + 2 \times log 7 - 4 \times log 3$ (this can’t be expressed in log 2 ,log 3)