Use logarithm properties to simplify numeric logarithmic expressions and estimate values.
Estimate which two integers a logarithm lies between using benchmark powers
Problem
Use consecutive powers of \(2\) to locate \(\log_{2}(0.2)\) between \(\text{two}~\text{consecutive}~\text{integers}\).
Big Picture
What this problem is really about
An argument between zero and one signals a negative logarithm when the base is greater than one. Compare it with consecutive negative powers of the base, remembering that the exponential function is still increasing across negative exponents. The order of the benchmark values therefore transfers directly to the exponent bounds.
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