Question

Evaluate and write each of the following in terms of x and y.if x= log 3 base 2 y= log 5 base 2.i.) log 7.5 base 2ii.) log 6750 base 2

Answer 1

$log_{2}(7.5) = log_{2}(15 \div 2) \\ = > log_{2}(15) - log_{2}(2) \\ = > log_{2}(5 \times 3) - 1 \\ = > log_{2}(5) + log_{2}(3) - 1 \\ = > y + x - 1$
$6750 = {5}^{3} \times {3}^{3} \times 2$
take logarithm with Base 2 in both sides
$log_{2}(6750) = log_{2}( {5}^{3} \times {3}^{3} \times 2) \\ = log_{2}( {5}^{3} ) + log_{2}( {3}^{3} ) + log_{2}(2) \\ = 3 log_{2}(5) + 3 log_{2}(3) + 1 \\ = 3y + 3x + 1$

Answer 2

Answer:

$log_2\,7.5=log_2\,3+log_2\,5-1=x+y-1$  and $log_2\,6750=3y+1+3x$

Step-by-step explanation:

Given, $x=log_2\,3\:\:and\:\:y=log_2\,5$

We use the following results,

$log_e\,e=1$

$log\,a\times b=log\,a+log\,b$

$log\,\frac{a}{b}=log\,a-log\,b$

$log\,x^a=a.log\,x$

i).

$log_2\,7.5=log_2\frac{75}{10}=log_2\,\frac{15}{2}=log_2\,15-log_2\,2$

$=log_2\,(3\times5)-1=log_2\,3+log_2\,5-1=x+y-1$

ii).

$log_2\,6750=log_2(5^3\times2\times3^3)=log_2\,5^3+log_2\,2+log_2\,3^3$

$=log_2\,5^3+1+log_2\,3^3=3.log_2\,5+1+3.log_2\,3=3y+1+3x$

Therefore, $log_2\,7.5=log_2\,3+log_2\,5-1=x+y-1$  and $log_2\,6750=3y+1+3x$