Question 92471
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If log<sub>b</sub>2 = x and log<sub>b</sub>3 = y, evaluate the following in terms of x and y:

You need these three rules:

1.     log<sub>b</sub>(ac) = log<sub>b</sub>a + log<sub>b</sub>c

2.     log<sub>b</sub>{{{a/c}}} = log<sub>b</sub>a - log<sub>b</sub>c

3.     log<sub>b</sub>{{{a^c}}} = c·log<sub>b</sub>a


(a) log<sub>b</sub>216=

Break 216 down to primes:

216 = {{{2*108}}} = {{{2*2*54}}} = {{{2*2*2*27}}} = {{{2*2*2*3*9}}} = {{{2*2*2*3*3*3}}} = {{{2^3*3^3}}}

log<sub>b</sub>216 = log<sub>b</sub>{{{2^3*3^3}}} 

= log<sub>b</sub>{{{2^3}}} + log<sub>b</sub>{{{3^3}}}     (by rule 1.)

= 3·log<sub>b</sub>2 + 3·log<sub>b</sub>3                 (by rule 3.)

= 3·x + 3·y                         (by substitution.) 

= 3(x + y) 

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(b) log<sub>b</sub>162=

Break 162 down to primes:

162 = {{{2*81}}} = {{{2*9*9}}} = {{{2*3*3*3*3}}} = {{{2*3^4}}}

log<sub>b</sub>162 = log<sub>b</sub>{{{2*3^4}}} 

= log<sub>b</sub>2 + log<sub>b</sub>{{{3^4}}}     (by rule 1.)

= log<sub>b</sub>2 + 4·log<sub>b</sub>3           (by rule 3.)

= x + 4·y                   (by substitution.) 

= x + 4y 

=======================================

(c) log<sub>b</sub>{{{2/9}}}

Break the 9 in the denominator down into primes

9 = {{{3*3}}} = {{{3^2}}}

log<sub>b</sub>{{{2/9}}} = log<sub>b</sub>{{{2/3^2}}}

= log<sub>b</sub>2 - log<sub>b</sub>{{{3^2}}}

= log<sub>b</sub>2 - 2·log<sub>b</sub>3

= x - 2y

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(d) 

 log<sub>b</sub>9
———————    
 log<sub>b</sub>8

Break the 9 down as {{{3^2}}} and the 8 down as {{{2^3}}}

 log<sub>b</sub>9     log<sub>b</sub>{{{3^2}}}
——————— = —————————    
 log<sub>b</sub>8     log<sub>b</sub>{{{2^3}}}


 2·log<sub>b</sub>3
—————————                (by rule 3.)    
 3·log<sub>b</sub>2

{{{(2y)/(3x)}}}            (substitution)

Edwin</pre>