Explanation:
A logarithm is an exponent. An integer exponent tells you the number of times the base is a factor in the product:
100,000 = 10×10×10×10×10 = 10⁵
so, log₁₀(100,000) = log₁₀(10⁵) = 5, the exponent of the base.
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You know that 10⁰ = 1 and 10¹ = 10. Since the logarithm is the exponent of 10, the log of a number between 1 and 10 is expected to be between 0 and 1.
You may also know that ...
[tex]10^{\frac{1}{2}}=\sqrt{10}\approx 3.1623[/tex]
That is, log₁₀(3.1623) = 0.50000. Just as the log of a number near 3 is between 0 and 1, the log₁₀ of any number between 1 and 10 will be between 0 and 1.
The number you're asking about, 5, has a logarithm that is about 0.69897.
