Two unrelated log parts: both solve equations

2

1. Use logarithms to solve the equation \(3 ^ { X } = 8\), giving your answer correct to 2 decimal places.
2. It is given that $$\ln z = \ln ( y + 2 ) - 2 \ln y$$ where \(y > 0\). Express \(z\) in terms of \(y\) in a form not involving logarithms.

10. (i) Use the laws of logarithms to solve the equation $$3 \log _ { 8 } 2 + \log _ { 8 } ( 7 - x ) = 2 + \log _ { 8 } x$$ (ii) Using algebra, find, in terms of logarithms, the exact value of \(y\) for which $$3 ^ { 2 y } + 3 ^ { y + 1 } = 10$$

11. (i) Given that \(x\) is a positive real number, solve the equation $$\log _ { x } 324 = 4$$ writing your answer as a simplified surd.
(ii) Given that $$\log _ { a } ( 5 y - 4 ) - \log _ { a } ( 2 y ) = 3 \quad y > 0.8,0 < a < 1$$ express \(y\) in terms of \(a\).

9. (i) Find the exact value of \(x\) for which $$2 \log _ { 10 } ( x - 2 ) - \log _ { 10 } ( x + 5 ) = 0$$ (ii) Given $$\log _ { p } ( 4 y + 1 ) - \log _ { p } ( 2 y - 2 ) = 1 \quad p > 2 , y > 1$$ express \(y\) in terms of \(p\).

9. (i) Find the exact value of \(x\) for which $$\log _ { 3 } ( x + 5 ) - 4 = \log _ { 3 } ( 2 x - 1 )$$ (ii) Given that $$3 ^ { y + 3 } \times 2 ^ { 1 - 2 y } = 108$$

1. show that $$0.75 ^ { y } = 2$$
2. Hence find the value of \(y\), giving your answer to 3 decimal places.

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6. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\). Give your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}, center]{0aafa21b-25f4-4f36-b914-bbaf6cae7a66-18_2674_1948_107_118}

5. (a) Find the positive value of \(x\) such that $$\log _ { x } 64 = 2$$ (b) Solve for \(x\) $$\log _ { 2 } ( 11 - 6 x ) = 2 \log _ { 2 } ( x - 1 ) + 3$$

8. (a) Find the value of \(y\) such that $$\log _ { 2 } y = - 3$$ (b) Find the values of \(x\) such that $$\frac { \log _ { 2 } 32 + \log _ { 2 } 16 } { \log _ { 2 } x } = \log _ { 2 } x$$

7. (i) Find the exact value of \(x\) for which $$\log _ { 2 } ( 2 x ) = \log _ { 2 } ( 5 x + 4 ) - 3$$ (ii) Given that $$\log _ { a } y + 3 \log _ { a } 2 = 5$$ express \(y\) in terms of \(a\).
Give your answer in its simplest form.

7. (i) \(2 \log ( x + a ) = \log \left( 16 a ^ { 6 } \right)\), where \(a\) is a positive constant Find \(x\) in terms of \(a\), giving your answer in its simplest form.
(ii) \(\quad \log _ { 3 } ( 9 y + b ) - \log _ { 3 } ( 2 y - b ) = 2\), where \(b\) is a positive constant Find \(y\) in terms of \(b\), giving your answer in its simplest form.

Find, giving your answer to 3 significant figures where appropriate, the value of \(x\) for which

1. \(3^x = 5\), [3]
2. \(\log_2(2x + 1) - \log_2 x = 2\). [4]

Solve

1. \(5^x = 8\), giving your answer to 3 significant figures, [3]
2. \(\log_2(x + 1) - \log_2 x = \log_2 7\). [3]

Solve each equation, giving your answers in exact form.

1. \(e^{4x-3} = 2\) [3]
2. \(\ln (2y - 1) = 1 + \ln (3 - y)\) [4]

In part (ii) of this question you must show detailed reasoning.

1. Use logarithms to solve the equation \(8^{2x+1} = 24\), giving your answer to 3 decimal places. [2]
2. Find the values of \(y\) such that $$\log_2(11y - 3) - \log_2 3 - 2\log_2 y = 1, \quad y > \frac{3}{11}$$ [6]

In this question you must show detailed reasoning. Solve the following equations.

1. \(y^6 + 7y^3 - 8 = 0\) [3]
2. \(9^{x+1} + 3^x = 8\) [4]

In this question you must show detailed reasoning. Solutions using calculator technology are not acceptable. Solve the following equations.

1. \(2\log_3(x + 1) = 1 + \log_3(x + 7)\) [4]
2. \(\log_y\left(\frac{1}{x}\right) = -\frac{3}{2}\) [3]