Express each of the following in exponential form :
(i) `log_(8) 0.125 = -1`
(ii) `log_(10) 0.01 = -2`
(iii) `log_(a) A = x`
(iv) `log_(10)1 = 0`

Answer

To express each of the given logarithmic equations in exponential form, we will use the property of logarithms which states that if \( \log_a b = c \), then it can be expressed as \( a^c = b \).

Let's solve each part step by step:

### (i) \( \log_{8} 0.125 = -1 \)

1. **Identify the base, the argument, and the result**: Here, the base \( a = 8 \), the argument \( b = 0.125 \), and the result \( c = -1 \).
2. **Apply the logarithmic identity**: According to the property, we can rewrite this as:
\[ 8^{-1} = 0.125 \]
3. **Final exponential form**: Thus, the exponential form is:
\[ 8^{-1} = 0.125 \]

### (ii) \( \log_{10} 0.01 = -2 \)

1. **Identify the base, the argument, and the result**: Here, \( a = 10 \), \( b = 0.01 \), and \( c = -2 \).
2. **Apply the logarithmic identity**: We can rewrite this as:
\[ 10^{-2} = 0.01 \]
3. **Final exponential form**: Thus, the exponential form is:
\[ 10^{-2} = 0.01 \]

### (iii) \( \log_{a} A = x \)

1. **Identify the base, the argument, and the result**: Here, \( a = a \), \( b = A \), and \( c = x \).
2. **Apply the logarithmic identity**: We can rewrite this as:
\[ a^{x} = A \]
3. **Final exponential form**: Thus, the exponential form is:
\[ a^{x} = A \]

### (iv) \( \log_{10} 1 = 0 \)

1. **Identify the base, the argument, and the result**: Here, \( a = 10 \), \( b = 1 \), and \( c = 0 \).
2. **Apply the logarithmic identity**: We can rewrite this as:
\[ 10^{0} = 1 \]
3. **Final exponential form**: Thus, the exponential form is:
\[ 10^{0} = 1 \]

### Summary of Exponential Forms:
1. \( 8^{-1} = 0.125 \)
2. \( 10^{-2} = 0.01 \)
3. \( a^{x} = A \)
4. \( 10^{0} = 1 \)