Changing bases

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Evaluate the following. Can you do this without calculating each factor separately?

1. \((\log_3 9)(\log_9 81)\)

2. \((\log_3 81)(\log_{81} 9)\)

Write the following as a single logarithm. What conditions must \(x\) and \(y\) satisfy?

1. \((\log_{10} 5)(\log_{5} x)\)

2. \((\log_{10} x)(\log_{x} y)(\log_{y} 3)\)

Simplify the following. What conditions must \(x\) and \(y\) satisfy?

1. \((\log_{x} y)(\log_{y} 10)(\log_{8} x)\)

2. \((\log_{x} y)(\log_{8} x)(\log_{y} 8)\)

The value of \(\log_2 3\) is approximately \(1.6\). Can you use this to approximate \(\log_4 27\)? What about \(\log_8 27?\) Can you generalise this result?

Evaluate

1. \(\dfrac{\log_{2} 5}{\log_{2} 25}\)

2. \(\dfrac{\log_{10} 5}{\log_{10} 3}\)

Given that \(\log_x y=4\) and \(\log_y 2 = 9\), evaluate

1. \((\log_x 2)(\log_y x)\)

2. \(\dfrac{\log_x 2}{\log_y x}\)

What is \((\log_a b^3)(\log_b a^3)\)?

What is the relationship between \(\log_a b\) and \(\log_b a\)?

Solve \(\log_{x} 7 = \log_{7} x\).