Logarithmic equation solving

4

1. Find the value of \(x\) satisfying the equation \(2 \ln ( x - 4 ) - \ln x = \ln 2\).
2. Use logarithms to find the smallest integer satisfying the inequality $$1.4 ^ { y } > 10 ^ { 10 }$$

3. Solve, giving each answer to 3 significant figures, the equations

1. \(4 ^ { a } = 20\)
2. \(3 + 2 \log _ { 2 } b = \log _ { 2 } ( 30 b )\) (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (i) Find, giving your answer to 3 significant figures, the value of \(y\) for which

$$3 ^ { y } = 12$$ (ii) Solve, giving an exact answer, the equation $$\log _ { 2 } ( x + 3 ) - \log _ { 2 } ( 2 x + 4 ) = 4$$ (You should show each step in your working.)

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

1. \(7 ^ { 2 x } = 14\)
2. \(\log _ { 5 } ( 3 x + 1 ) = - 2\)

13. (i) Find the value of \(x\) for which $$4 ^ { 3 x + 2 } = 3 ^ { 600 }$$ giving your answer to 4 significant figures.
(ii) Given that $$\log _ { a } ( 3 b - 2 ) - 2 \log _ { a } 5 = 4 , \quad a > 0 , a \neq 1 , b > \frac { 2 } { 3 }$$ find an expression for \(b\) in terms of \(a\).

3. (i) Solve $$7 ^ { x + 2 } = 3$$ giving your answer in the form \(x = \log _ { 7 } a\) where \(a\) is a rational number in its simplest form.
(ii) Using the laws of logarithms, solve $$1 + \log _ { 2 } y + \log _ { 2 } ( y + 4 ) = \log _ { 2 } ( 5 - y )$$

8. (i) Find the exact solution of the equation $$8 ^ { 2 x + 1 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.
(ii) Using the laws of logarithms, solve $$\log _ { 5 } ( 7 - 2 y ) = 2 \log _ { 5 } ( y + 1 ) - 1$$

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve $$3 ^ { a } = 70$$ giving the answer to 3 decimal places.
2. Find the exact value of \(b\) such that $$4 + 3 \log _ { 3 } b = \log _ { 3 } 5 b$$

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

1. \(3 ^ { x } = 5\),
2. \(\log _ { 2 } ( 2 x + 1 ) - \log _ { 2 } x = 2\).

6. (a) Find, to 3 significant figures, the value of \(x\) for which \(8 ^ { x } = 0.8\).
(b) Solve the equation $$2 \log _ { 3 } x - \log _ { 3 } 7 x = 1$$

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

1. \(5 ^ { x } = 10\),
2. \(\log _ { 3 } ( x - 2 ) = - 1\).

8. (i) Solve $$5 ^ { y } = 8$$ giving your answer to 3 significant figures.
(ii) Use algebra to find the values of \(x\) for which $$\log _ { 2 } ( x + 15 ) - 4 = \frac { 1 } { 2 } \log _ { 2 } x$$

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

8. (i) Given that $$\log _ { 3 } ( 3 b + 1 ) - \log _ { 3 } ( a - 2 ) = - 1 , \quad a > 2$$ express \(b\) in terms of \(a\).
(ii) Solve the equation $$2 ^ { 2 x + 5 } - 7 \left( 2 ^ { x } \right) = 0$$ giving your answer to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

7. (i) Find the value of \(y\) for which $$1.01 ^ { y - 1 } = 500$$ Give your answer to 2 decimal places.
(ii) Given that $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$

1. show that $$9 x ^ { 2 } + 18 x - 7 = 0$$
2. Hence solve the equation $$2 \log _ { 4 } ( 3 x + 5 ) = \log _ { 4 } ( 3 x + 8 ) + 1 , \quad x > - \frac { 5 } { 3 }$$ DO NOTI WRITE IN THIS AREA

Solve

1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).

8

1. Solve the equation \(10 ^ { x } = 316\).
2. Simplify \(\log _ { a } \left( a ^ { 2 } \right) - 4 \log _ { a } \left( \frac { 1 } { a } \right)\).

5. (i) Solve the equation $$\log _ { 2 } ( 6 - x ) = 3 - \log _ { 2 } x$$ (ii) Find the smallest integer \(n\) such that $$3 ^ { n - 2 } > 8 ^ { 250 }$$

5. (a) Given that \(t = \log _ { 3 } x\),

1. write down an expression in terms of \(t\) for \(\log _ { 3 } x ^ { 2 }\),
2. show that \(\log _ { 9 } x = \frac { 1 } { 2 } t\).
   (b) Hence, or otherwise, find to 3 significant figures the value of \(x\) such that $$\log _ { 3 } x ^ { 2 } - \log _ { 9 } x = 4$$

4. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for

1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
2. \(\quad \log _ { 2 } ( \sqrt { x } )\).
   (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$

5. (i) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (ii) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.

3. (a) Given that \(y = \ln x\),

1. find an expression for \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\) in terms of \(y\),
2. show that \(\log _ { 2 } x = \frac { y } { \ln 2 }\).
   (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.

3

1. Express \(2 \ln x + \ln 3\) as a single logarithm.
2. Hence, given that \(x\) satisfies the equation $$2 \ln x + \ln 3 = \ln ( 5 x + 2 )$$ show that \(x\) is a root of the quadratic equation \(3 x ^ { 2 } - 5 x - 2 = 0\).
3. Solve this quadratic equation, explaining why only one root is a valid solution of $$2 \ln x + \ln 3 = \ln ( 5 x + 2 ) .$$

1．（a）By writing \(u = \log _ { 4 } r\) ，where \(r > 0\) ，show that $$\log _ { 4 } r = \frac { 1 } { 2 } \log _ { 2 } r$$ （b）Solve the equation $$\log _ { 4 } \left( 5 x ^ { 2 } - 11 \right) = \log _ { 2 } ( 3 x - 5 )$$

4

1. Use logarithms to solve the equation \(5 ^ { x - 1 } = 120\), giving your answer correct to 3 significant figures.
2. Solve the equation \(\log _ { 2 } x + 2 \log _ { 2 } 3 = \log _ { 2 } ( x + 5 )\).