Exponentials and logarithms: Practice Questions

Solve the equation \(\log_2(x + 3) + \log_2(x - 3) = 4\) for \(x > 3\).

Solve the equation \(2^{2x+1} - 17(2^x) + 8 = 0\).

Find the exact solution to the equation \(3^{x+1} = 5^x\) in terms of natural logarithms.

A graph of \(\log_{10} y\) is plotted against \(\log_{10} x\). The resulting line is straight and passes through the points \((0, 2)\) and \((3, 8)\). Express \(y\) in terms of \(x\).

Solve the simultaneous equations:
\(\log_y x = 2\)
\(xy = 27\)

Given that \(\log_a 4 + \log_a 10 = 3\), find the exact value of \(a\), giving your answer in the form \(k\sqrt[3]{5}\) where \(k\) is an integer.

The value of a specialized industrial machine, \(V\) pounds, is modeled by the equation \(V = 25000e^{-kt}\), where \(t\) is the time in years since the machine was purchased and \(k\) is a positive constant.
(a) Given that the machine is worth £15,000 after 4 years, calculate the value of \(k\) to 3 decimal places.
(b) Find the age of the machine when its value has decreased to £7,500.
(c) Explain why the value of the machine will never reach zero according to this model.