Express variable in terms of another

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.

2 It is given that \(\ln ( y + 1 ) - \ln y = 1 + 3 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

1 Given that \(2 \ln ( x + 4 ) - \ln x = \ln ( x + a )\), express \(x\) in terms of \(a\).

1 It is given that \(z = \ln ( y + 2 ) - \ln ( y + 1 )\). Express \(y\) in terms of \(z\).

1 Given that \(\ln \left( 1 + \mathrm { e } ^ { 2 y } \right) = x\), express \(y\) in terms of \(x\).

2 It is given that \(\ln ( y + 5 ) - \ln y = 2 \ln x\). Express \(y\) in terms of \(x\), in a form not involving logarithms.

1 It is given that \(x = \ln ( 2 y - 3 ) - \ln ( y + 4 )\).
Express \(y\) in terms of \(x\).

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 \(p\).

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}

6. Given that $$\log _ { x } ( 7 y + 1 ) - \log _ { x } ( 2 y ) = 1 , \quad x > 4 , \quad 0 < y < 1$$ express \(y\) in terms of \(x\).

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.

7

1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
   (a) \(\log _ { 10 } \left( \frac { x } { y } \right)\) (b) \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
2. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.

1. Given that \(a > b > 0\) and that \(a\) and \(b\) satisfy the equation

$$\log a - \log b = \log ( a - b )$$

1. show that $$a = \frac { b ^ { 2 } } { b - 1 }$$ (3)
2. Write down the full restriction on the value of \(b\), explaining the reason for this restriction.

4. Given that \(a\) is a positive constant and $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7$$ show that \(a = \ln k\), where \(k\) is a constant to be found.

2

1. Express \(2 \log _ { 3 } x + \log _ { 3 } a\) as a single logarithm.
2. Given that \(2 \log _ { 3 } x + \log _ { 3 } a = 2\), express \(x\) in terms of \(a\).

7

1. Given that $$\log _ { a } x = \log _ { a } 16 - \log _ { a } 2$$ write down the value of \(x\).
2. Given that $$\log _ { a } y = 2 \log _ { a } 3 + \log _ { a } 4 + 1$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.

8

1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}

7

1. Sketch the graph of \(y = \frac { 1 } { 2 ^ { x } }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(\frac { 1 } { 2 ^ { x } } = \frac { 5 } { 4 }\), giving your answer to three significant figures.
3. Given that $$\log _ { a } \left( b ^ { 2 } \right) + 3 \log _ { a } y = 3 + 2 \log _ { a } \left( \frac { y } { a } \right)$$ express \(y\) in terms of \(a\) and \(b\).
   Give your answer in a form not involving logarithms.

4 Given that $$\log _ { a } N - \log _ { a } x = \frac { 3 } { 2 }$$ express \(x\) in terms of \(a\) and \(N\), giving your answer in a form not involving logarithms.
(3 marks)

5

1. Given that $$\log _ { a } x = 2 \log _ { a } 6 - \log _ { a } 3$$ show that \(x = 12\).
2. Given that $$\log _ { a } y + \log _ { a } 5 = 7$$ express \(y\) in terms of \(a\), giving your answer in a form not involving logarithms.
   (3 marks)

7

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

9 Given that $$\log _ { 2 } x ^ { 3 } - \log _ { 2 } y ^ { 2 } = 9$$ show that $$x = A y ^ { p }$$ where \(A\) is an integer and \(p\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-15_2488_1716_219_153}