Evaluate log expression using laws

7

1. Evaluate \(\log _ { 5 } 15 + \log _ { 5 } 20 - \log _ { 5 } 12\).
2. Given that \(y = 3 \times 10 ^ { 2 x }\), show that \(x = a \log _ { 10 } ( b y )\), where the values of the constants \(a\) and \(b\) are to be found.

9 Simplify

1. \(10 - 3 \log _ { a } a\),
2. \(\frac { \log _ { 10 } a ^ { 5 } + \log _ { 10 } \sqrt { a } } { \log _ { 10 } a }\). Section B (36 marks)

7 Simplify

1. \(\log _ { 10 } x ^ { 5 } + 3 \log _ { 10 } x ^ { 4 }\),
2. \(\log _ { a } 1 - \log _ { a } a ^ { b }\).

9 You are given that $$\log _ { a } x = \frac { 1 } { 2 } \log _ { a } 16 + \log _ { a } 75 - 2 \log _ { a } 5 .$$ Find the value of \(x\).

5

1. Given that $$\log _ { a } x = 3 \log _ { a } 6 - \log _ { a } 8$$ where \(a\) is a positive constant, show that \(x = 27\).
2. Write down the value of:
   1. \(\quad \log _ { 4 } 1\);
   2. \(\log _ { 4 } 4\);
   3. \(\log _ { 4 } 2\);
   4. \(\quad \log _ { 4 } 8\).

5

1. Write down the value of:
   1. \(\log _ { a } 1\);
   2. \(\log _ { a } a\).
2. Given that $$\log _ { a } x = \log _ { a } 5 + \log _ { a } 6 - \log _ { a } 1.5$$ find the value of \(x\).

1 Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer.
[0pt] [1 mark] $$- 2 \log _ { 10 } \left( \frac { 1 } { a } \right) \quad 2 \log _ { 10 } ( a ) \quad \log _ { 10 } \left( a ^ { 2 } \right) \quad - 4 \log _ { 10 } ( \sqrt { a } )$$