Combine logs into single logarithm

3 Express each of the following as a single logarithm:

1. \(\log _ { a } 2 + \log _ { a } 3\),
2. \(2 \log _ { 10 } x - 3 \log _ { 10 } y\).

3

1. Write \(\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 )\) as a single logarithm.
2. Without using your calculator, verify that \(x = 4\) is a root of the equation $$\log _ { 10 } ( x + 4 ) - 2 \log _ { 10 } x + \log _ { 10 } ( x + 16 ) = 1$$

6

1. Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
   1. \(\log _ { a } 4 + \log _ { a } 10\);
   2. \(\log _ { a } 16 - \log _ { a } 2\);
   3. \(3 \log _ { a } 5\).
2. Use logarithms to solve the equation \(( 1.5 ) ^ { 3 x } = 7.5\), giving your value of \(x\) to three decimal places.
3. Given that \(\log _ { 2 } p = m\) and \(\log _ { 8 } q = n\), express \(p q\) in the form \(2 ^ { y }\), where \(y\) is an expression in \(m\) and \(n\).

1 Express as a single logarithm $$\log _ { 10 } 2 - \log _ { 10 } x$$ Circle your answer.
[0pt] [1 mark] \(\log _ { 10 } ( 2 + x ) \quad \log _ { 10 } ( 2 - x ) \quad \log _ { 10 } ( 2 x ) \quad \log _ { 10 } \left( \frac { 2 } { x } \right)\)

3 Express as a single logarithm $$2 \log _ { a } 6 - \log _ { a } 3$$