Question

Why does log 10 100,000 equal to 5?

Answer 1

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 ...

  $10^{\frac{1}{2}}=\sqrt{10}\approx 3.1623$

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.