1. Find the following logarithms without using a calculator:

2. Express in terms of ln x and ln y:

3. Express as a single logarithm:

4. Use the change of base formula to simplify:

5. Simplify the following:

6. Use the substitution u = ex to reduce the equation e^2x − 3e^x + 2 = 0 to a quadratic equation in u. Factorise this quadratic and hence find the value(s) of u which satisfy it. What are the value(s) of x, correct to two decimal places, which satisfy the original equation?

7. A liquid which has been heated was allowed to cool down in a room where the ambient temperature of the air remained at 10.4°C. The law of cooling yields a relationship of the form T = b + ae^λt where T °C is the temperature of the liquid at time t (minutes) and a, b and λ are parameters. Why must b = 10.4? What can you deduce about the values (relative to zero) of a and λ? Values of T are given for particular values of t in the following table:

Plot a graph of T against t. Let y = ln(T − b) and draw up a new table of y against the values of t given above. Plot a second graph, of y against t, and, by eye, draw the ‘best’ fitting straight line through your points. Estimate the values of a and λ, and so deduce the initial temperature of the liquid.

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(a) log 8 (d) log, 81 (b) log, (e) log, 3 (c) log 1 (f) log, 0.5