Express each of the following in logarithmic form :

(i) 5³ = 125

(ii) 3^-2 = $\dfrac{1}{9}$

(iii) 10^-3 = 0.001

(iv) $(81)^{\dfrac{3}{4}} = 27$

(i) Given,

⇒ 5³ = 125

⇒ log₅125 = 3.

Hence, required logarithmic form is log₅ 125 = 3.

(ii) Given,

$\Rightarrow 3^{-2} = \dfrac{1}{9} \\[1em] \Rightarrow \text{log}_{3} \space {\dfrac{1}{9}} = -2.$

Hence, required logarithmic form is $\text{log}_{3} \space {\dfrac{1}{9}} = -2.$

(iii) Given,

⇒ 10^-3 = 0.001

⇒ log₁₀ 0.001 = -3.

Hence, required logarithmic form is log₁₀ 0.001 = -3.

(iv) Given,

$\Rightarrow (81)^{\dfrac{3}{4}} = 27 \\[1em] \Rightarrow \text{log}_{81} \space 27 = \dfrac{3}{4}.$

Hence, required logarithmic form is $\text{log}_{81} \space 27 = \dfrac{3}{4}.$