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6
4 (a) Given
log_(2)3=p and
log_(2)5=q, express and simplify
log_(8)((2)/(5))+log_(3)2 in terms of
p and
q.
[3]

$6$ $\newline$ $4$ (a) Given $\log _{2} 3=p$ and $\log _{2} 5=q$ , express and simplify $\log _{8} \frac{2}{5}+\log _{3} 2$ in terms of $p$ and $q$ . $\newline$ [ $3$ ]

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Quotient property of logarithms

Full solution

Q.

6

\newline

4

(a) Given

\log _{2} 3=p

and

\log _{2} 5=q

, express and simplify

\log _{8} \frac{2}{5}+\log _{3} 2

in terms of

p

and

q

.

\newline

[

3

]

Change Base to Base $2$ : Change the base of $\log_{8}$ to base $2$ using the change of base formula: $\log_{b} a = \frac{\log_{k} a}{\log_{k} b}$ .
$\log_{8} \left(\frac{2}{5}\right) = \frac{\log_{2} \left(\frac{2}{5}\right)}{\log_{2} 8}.$
Simplify Log Base $2$ of $8$ : Simplify $\log_{2} 8$ since $8$ is $2$ cubed, $\log_{2} 8$ equals $3$ . $\newline$ $\log_{8} \left(\frac{2}{5}\right) = \frac{\log_{2} \left(\frac{2}{5}\right)}{3}.$
Split Log Base $2$ of $(\frac{2}{5})$ : Split log base $2$ of $(\frac{2}{5})$ into log base $2$ of $2$ minus log base $2$ of $5$ using the quotient rule. $\newline$ $\log_{8}(\frac{2}{5}) = (\log_{2} 2 - \log_{2} 5) / 3$ .
Substitute Given Values: Substitute the given values: $\log_{2} 3 = p$ and $\log_{2} 5 = q$ . $\newline$ $\log_{8} \left(\frac{2}{5}\right) = \frac{1 - q}{3}$ because $\log_{2} 2 = 1$ .
Change Base to Base $2$ : Change the base of $\log_{3} 2$ to base $2$ using the change of base formula. $\log_{3} 2 = \frac{\log_{2} 2}{\log_{2} 3}$ .
Substitute Given Value: Substitute the given value for $\log_{2} 3$ . $\newline$ $\log_{3} 2 = \frac{1}{p}$ because $\log_{2} 2 = 1$ .
Combine Expressions: Combine the two expressions. $\newline$ $\log_{8}\left(\frac{2}{5}\right) + \log_{3}2 = \frac{1 - q}{3} + \frac{1}{p}$ .
Simplify Expression: Simplify the expression. $\newline$ Final expression in terms of $p$ and $q$ : $\frac{1 - q}{3} + \frac{1}{p}$ .

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