Explanation :

$~~~\dfrac{\log_{\text{a}}{\text{n}}}{\log_{\text{ab}}{\text{n}}}$

= $\dfrac {\log \text{n}}{\log \text{a}} \times \dfrac{\log \text{ab}}{\log\text{n}}$

= $\dfrac {\log \text{ab}} {\log \text{a}}$

= $\dfrac{\log \text{a} + \log \text{b}} {\log \text{a}}$

= $\dfrac {\log \text{a}}{\log \text{a}} + \dfrac {\log \text{b}}{\log \text{a}}$

= $1 + \log_{\text{a}}{\text{b}}$

Explanation :

∴ Given expression = 1 + 2 + 3 + 4 + 5 = 15

Explanation :

∵ log₈ x + log₄ x + log₂ x = 11

⇒ log₂³ x + log₂² x + log₂ x = 11

⇒ $\dfrac{1}{3}$ log₂ x + $\dfrac{1}{2}$ log₂ x + log₂ x = 11

⇒ $\left(\dfrac{1}{3} + \dfrac{1}{2}+1\right)$ log₂x = 11

⇒ $\dfrac{11}{6}$ log₂ x = 11

⇒ log₂ x = 11 x $\dfrac{6}{11}$

⇒ log₂ x = 6

⇒ x = 2⁶

⇒ x = 64

Explanation :

$~~~ \dfrac{1}{\log_{({\text{p/q}})}\text{x}} + \dfrac{1}{\log_{({\text{q/r}})}\text{x}} + \dfrac{1}{\log_{({\text{r/p}})}\text{x}} $

= ${\log_{\text{x}}\left( \dfrac{\text{p}}{\text{q}} \right)} + {\log_{\text{x}}\left( \dfrac{\text{q}}{\text{r}} \right)} + {\log_{\text{x}}\left( \dfrac{\text{r}}{\text{p}} \right)}$

= $\log_{\text{x}}\left( \dfrac{\text{p}}{\text{q}} \times \dfrac{\text{q}}{\text{r}} \times \dfrac{\text{r}}{\text{p}} \right)$

= $\log_{\text{x}}{1}$

= 0

Explanation :

$~~~\dfrac{1}{\log_{\text a}(\text {bc}) + 1}$ + $\dfrac{1}{\log_{\text b}(\text {ac}) + 1}$ + $\dfrac{1}{ \log_{\text c} (\text {ab}) + 1}$

= $\dfrac{1}{\log_{\text a}(\text {bc}) + \log_{\text a}{\text a}}$ + $\dfrac{1}{\log_{\text b}(\text {ac}) + \log_{\text b}{\text b}}$ + $\dfrac{1}{\log_{\text c}(\text {ab}) + \log_{\text c}{\text c}}$

= $\dfrac{1}{\log_{\text a}\left( \text {abc} \right)}$ +  $\dfrac{1}{\log_{\text b}\left( \text {abc} \right)}$ +  $\dfrac{1}{\log_{\text c}\left( \text{abc} \right)}$

= $\log_{\text{abc}}{\text {a}}$+ $\log_{\text{abc}}{\text {b}}$ + $\log_{\text{abc}}{\text {c}}$

= $\log_{\text{abc}}{\text {abc}}$

= 1