Express each of the following in logarithmic form :
(i) `5^(3) = 125`
(ii) `3^(-2) = (1)/(9)`
(iii) `10^(-3) = 0.001`
(iv) `(81)^((3)/(4)) = 27`

Answer

To express each of the given equations in logarithmic form, we will follow the general rule that if \( a^b = c \), then it can be expressed as \( \log_a(c) = b \).

### Solution:

**(i)** Given: \( 5^3 = 125 \)

1. Identify the base \( a = 5 \), the exponent \( b = 3 \), and the result \( c = 125 \).
2. Using the logarithmic form:
\[ \log_5(125) = 3 \]

**(ii)** Given: \( 3^{-2} = \frac{1}{9} \)

1. Identify the base \( a = 3 \), the exponent \( b = -2 \), and the result \( c = \frac{1}{9} \).
2. Using the logarithmic form:
\[ \log_3\left(\frac{1}{9}\right) = -2 \]

**(iii)** Given: \( 10^{-3} = 0.001 \)

1. Identify the base \( a = 10 \), the exponent \( b = -3 \), and the result \( c = 0.001 \).
2. Using the logarithmic form:
\[ \log_{10}(0.001) = -3 \]

**(iv)** Given: \( 81^{\frac{3}{4}} = 27 \)

1. Identify the base \( a = 81 \), the exponent \( b = \frac{3}{4} \), and the result \( c = 27 \).
2. Using the logarithmic form:
\[ \log_{81}(27) = \frac{3}{4} \]

### Final Answers:
1. \( \log_5(125) = 3 \)
2. \( \log_3\left(\frac{1}{9}\right) = -2 \)
3. \( \log_{10}(0.001) = -3 \)
4. \( \log_{81}(27) = \frac{3}{4} \)