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 \(x\).

2 You are given that \(\ln ( 12 ) = 2.484907\) and \(\ln ( 3 ) = 1.098612\), correct to 6 decimal places. Use the laws of logarithms to obtain the values of \(\ln ( 36 )\) and \(\ln ( 0.5 )\), correct to 4 decimal places. You must show your numerical working.

Find the value of \(\log_a(a^5) + \log_a\left(\frac{1}{a}\right)\) [2 marks]