Logarithmic equation solving

8

1. Use logarithms to solve the equation \(7 ^ { w - 3 } - 4 = 180\), giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$

8

1. Use logarithms to solve the equation \(5 ^ { 3 w - 1 } = 4 ^ { 250 }\), giving the value of \(w\) correct to 3 significant figures.
2. Given that \(\log _ { x } ( 5 y + 1 ) - \log _ { x } 3 = 4\), express \(y\) in terms of \(x\).

8

1. Use logarithms to solve the equation $$2 ^ { n - 3 } = 18000$$ giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 2 } x + \log _ { 2 } y = 8 , \quad \log _ { 2 } \left( \frac { x ^ { 2 } } { y } \right) = 7$$

5

1. The graph of \(y = 2 ^ { x }\) can be transformed to the graph of \(y = 2 ^ { x + 4 }\) either by a translation or by a stretch.
   1. Give full details of the translation.
   2. Give full details of the stretch.
2. In this question you must show detailed reasoning. Solve the equation \(\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )\).

5 In 2019 scientists developed a model for comparing the ages of humans and dogs.
According to the model, \(Y = A \ln X + B\) where \(X =\) dog age in years and \(Y =\) human age in years.
For the model, it is known that when \(X = 1 , Y = 31\) and when \(X = 12 , Y = 71\).

1. Find the value of \(B\).
2. Determine the value of \(A\), correct to the nearest whole number. Use the model, with the exact value of \(B\) and the value of \(A\) correct to the nearest whole number, to answer parts (c) and (d).
3. Find the human age corresponding to a dog age of 20 years.
4. Determine the dog age corresponding to a human age of 120 years.

9

1. 1. Find the value of \(p\) for which \(\sqrt { 125 } = 5 ^ { p }\).
   2. Hence solve the equation \(5 ^ { 2 x } = \sqrt { 125 }\).
2. Use logarithms to solve the equation \(3 ^ { 2 x - 1 } = 0.05\), giving your value of \(x\) to four decimal places.
3. It is given that $$\log _ { a } x = 2 \left( \log _ { a } 3 + \log _ { a } 2 \right) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms.
   (4 marks)

9

1. Use logarithms to solve the equation \(2 ^ { 3 x } = 5\), giving your value of \(x\) to three significant figures.
2. Given that \(\log _ { a } k - \log _ { a } 2 = \frac { 2 } { 3 }\), express \(a\) in terms of \(k\).
3. 1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 3 }\) in ascending powers of \(x\).
   2. It is given that $$\log _ { 2 } \left[ ( 1 + 2 n ) ^ { 3 } - 8 n \right] = \log _ { 2 } ( 1 + 2 n ) + \log _ { 2 } \left[ 4 \left( 1 + n ^ { 2 } \right) \right]$$ By forming and solving a suitable quadratic equation, find the possible values of \(n\). [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-20_1581_1714_1126_153} \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-24_2488_1728_219_141}

4. Solve each equation, giving your answers to an appropriate degree of accuracy.

1. \(3 ^ { x - 2 } = 5\)
2. \(\quad \log _ { 2 } ( 6 - y ) = 3 - \log _ { 2 } y\)

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