Question

If loga2 = x and loga 5 = y, find in terms of x and y, expressions for
(a) log2 ^5;
(b) loga ^20.

Answer 1

$\textsf{\large{\underline{Solution}:}}$

Given That:

$\rm: \longmapsto log_{a}(2) = x$

$\rm: \longmapsto log_{a}(5) = y$

For question a:

We know that:

$\rm: \longmapsto log_{x}(y) = \dfrac{ log_{a}(y) }{ log_{a}(x) } ,x > 0, y>0,a>0, x\ne 1,a\ne 1$

Therefore:

$\rm: \longmapsto log_{2}(5) = \dfrac{ log_{a}(5) }{ log_{a}(2) }$

$\rm: \longmapsto log_{2}(5) = \dfrac{y}{x}$

Which is our required answer.

For question b:

We have:

$\rm = log_{a}(20)$

$\rm = log_{a}(4 \times 5)$

$\rm = log_{a}(4) + log_{a}(5)$

$\rm = log_{a}( {2}^{2} ) + log_{a}(5)$

$\rm = 2 \: log_{a}(2) + log_{a}(5)$

$\rm = 2x + y$

$\rm: \longmapsto log_{a}(20) = 2x + y$

Which is our required answer.

$\textsf{\large{\underline{More To Know}:}}$

$\rm 1. \: \: {a}^{n} = b \implies log_{a}(b) = n$

$\rm 2. \: \: log_{a}(1) = 0, \: a \neq0,1$

$\rm 3. \: \: log_{a}(a) = 1, \: a \neq0,1$

$\rm 4. \: \: log_{a}(x) = log_{a}(y) \implies x = y$

$\rm 5. \: \: log_{e}(x) = ln(x)$

$\rm6. \: \: log_{a}(x) + log_{a}(y) = log_{a}(xy)$

$\rm7. \: \: log_{a}(x) - log_{a}(y) = log_{a} \bigg( \dfrac{x}{y} \bigg)$

$\rm 8. \: \: log_{a}( {x}^{n} ) = n\log_{a}(x)$

$\rm 9. \: \: log_{a}(m) = \dfrac{ log_{b}(m) }{ log_{b}(a) },m > 0,b > 0,a \ne1,b \ne1$

$\rm 10. \: \: log_{a}(b) = \dfrac{1}{ log_{b}(a) }$

Answer 2

Step-by-step explanation:

given :

• If loga2 = x and loga 5 = y, find in terms of x and y, expressions for

to find :

• find in terms of x and y, expressions for

solution :

• = loga (20)

• = loga (4 x 5)

• = loga (4) + loga (5)

• loga (2²) + loga (5)

• = 2 loga (2) + loga (5)

• = 2x + y

• :loga (20) = 2x + y