1. Express each of the following in logarithmic form:
(i) \( 10^3 = 1000 \)
\[\log_{10} 1000 = 3\]
(ii) \( 2^8 = 256 \)
\[\log_{2} 256 = 8\]
(iii) \( 3^{-3} = \frac{1}{27} \)
\[\log_{3} \frac{1}{27} = -3\]
(iv) \( 20^2 = 400 \)
\[\log_{20} 400 = 2\]
(v) \( 16^{-\frac{1}{4}} = \frac{1}{2} \)
\[\log_{16} \frac{1}{2} = -\frac{1}{4}\]
(vi) \( 11^2 = 121 \)
\[\log_{11} 121 = 2\]
(vii) \( p = q^r \)
\[\log_{q} p = r\]
(viii) \( (32)^{\frac{-1}{5}} = \frac{1}{2} \)
\[\log_{32} \frac{1}{2} = \frac{-1}{5}\]
2. Express each of the following in exponential form:
(i) \( \log_{5} 125 = 3 \)
\[5^3 = 125\]
(ii) \( \log_{2} 16 = 4 \)
\[2^4 = 16\]
(iii) \( \log_{23} 1 = 0 \)
\[23^0 = 1\]
(iv) \( \log_{5} 5 = 1 \)
\[5^1 = 5\]
(v) \( \log_{2} \frac{1}{8} = -3 \)
\[2^{-3} = \frac{1}{8}\]
(vi) \( \frac{1}{2} = \log_{9} 3 \)
\[9^{\frac{1}{2}} = 3\]
(vii) \( 5 = \log_{10} 100000 \)
\[10^5 = 100000\]
(viii) \( \log_{4} \frac{1}{16} = -2 \)
\[4^{-2} = \frac{1}{16}\]
3. Find the value of \( x \) in each of the following:
(i) \( \log_{x} 64 = 3 \)
\[x^3 = 64\]
\[x = 4\]
(ii) \( \log_{5} 1 = x \)
\[5^x = 1\]
\[x = 0\]
(iii) \( \log_{x} 8 = 1 \)
\[x^1 = 8\]
\[x = 8\]
(iv) \( \log_{10} x = -3 \)
\[10^{-3} = x\]
\[x = 0.001\]
(v) \( \log_{4} x = \frac{3}{2} \)
\[4^{\frac{3}{2}} = x\]
\[x = 8\]
(vi) \( \log_{2} 1024 = x \)
\[2^x = 1024\]
\[x = 10\]
