Question

Explain the log formula ​

Answer 1

Explanation:

The log formula rearranges an exponential term. I find it helpful to remember that a logarithm is an exponent.

If you start with the exponential term ...

  $b^x=a$

you will notice that the base of it is "b" and the exponent is "x". The log formula rearranges this to ...

  $\log_b{(a)}=x$

That is, "x" is the power to which the base must be raised in order to give you the value "a".

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Examples

  $10^3=1000\\\\\log{(1000)}=3 \qquad\text{the base of 10 is assumed}$

When the base is e ≈ 2.7182818284590..., the logarithm is called the "natural log". Its function name is differentiated from the log base 10 by calling it "ln".

  $e^{-0.693147}\approx 0.5\\\\\ln{(0.5)}\approx -0.693147$

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You know that multiplication is performed by adding exponents of powers of the same base:

  $10^5\times 10^3=10^{5+3}=10^8$

Logarithms work the same way that exponent arithmetic works:

  $\log{(10^5\times 10^3)}=\log{(10^5)}+\log{(10^3)}\\\\=5+3=8=\log{(10^8)}$

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e is an irrational number. One of the possible definitions of it is shown in the attachment.