Converting Logarithms Between Bases

00:00 3 marks

Using the change-of-base formula, express \(\log_{10}(x)\) in terms of the natural logarithm \(\ln(x)\). Then, given that \(\ln(2) \approx 0.693\) and \(\ln(10) \approx 2.303\), calculate an approximate value for \(\log_{10}(2)\). Show your working clearly.

Hint Hint

Hint

Recall the change-of-base formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\). To express in terms of natural logarithm, use base \(c=e\).

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Complete Answer

\(\log_{10}(x) = \frac{\ln(x)}{\ln(10)}\); \(\log_{10}(2) \approx \frac{0.693}{2.303} \approx 0.301\)

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