4.5.1 Indices and Logarithms, Short Questions (Question 1 - 4)

Question 1

Solve the equation, log₃ [log₂(2x – 1)] = 2

Solution:

log₃ [log₂ (2x – 1)] = 2 ← (if log _aN = x, N = a^x)

log₂ (2x – 1) = 3²
log₂ (2x – 1) = 9

2x – 1 = 2⁹

x = 256.5

Question 2

Solve the equation,

l o g_{16} [l o g_{2} (5 x - 4)] = l o g_{9} \sqrt{3}

Solution:

\begin{array}{l} l o g_{16} [l o g_{2} (5 x - 4)] = l o g_{9} \sqrt{3} \\ l o g_{16} [l o g_{2} (5 x - 4)] = \frac{1}{4} \leftarrow \begin{array}{l} \log_{9} \sqrt{3} = \log_{9} 3^{\frac{1}{2}} = \frac{1}{2} \log_{9} 3 \\ = \frac{1}{2} (\frac{1}{\log_{3} 9}) = \frac{1}{2} (\frac{1}{2}) = \frac{1}{4} \end{array} \\ l o g_{2} (5 x - 4) = 16^{\frac{1}{4}} \\ l o g_{2} (5 x - 4) = 2 \\ 5 x - 4 = 2^{2} \\ 5 x = 8 \\ x = \frac{8}{5} \end{array}

Question 3

Solve the equation,

5^{\log_{4} x} = 125

Solution:

\begin{array}{l} 5^{\log_{4} x} = 125 \\ \log_{5} 5^{\log_{4} x} = \log_{5} 125 \leftarrow put log for both side \\ (\log_{4} x) (\log_{5} 5) = 3 \\ (\log_{4} x) (1) = 3 \\ x = 4^{3} = 64 \end{array}

Question 4

Solve the equation,

5^{\log_{5} (x + 1)} = 9

Solution:

\begin{array}{l} 5^{\log_{5} (x + 1)} = 9 \\ \log_{5} 5^{^{\log_{5} (x + 1)}} = \log_{5} 9 \\ \log_{5} (x + 1) . \log_{5} 5 = \log_{5} 9 \\ \log_{5} (x + 1) = \log_{5} 9 \\ x + 1 = 9 \\ x = 8 \end{array}

4.5.1 Indices and Logarithms, Short Questions (Question 1 – 4)