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How does a logarithm followed by a variable read such as ($\ln x$) or ($\log x$). Is it $\log$ times $x$ or the $\log$ of $x$? I'm a little confused by this...?

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"Log" is a function; hence, interpreting $\log x$ as "$\log$ times $x$" doesn't make any sense - $\log$ needs an input before it can be interpreted as a number. The notations $\log x$ and $\ln x$ mean the exact same thing as $\log(x)$ and $\ln(x)$; they are just used as short-hand when leaving off the parentheses will not cause confusion.

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Thank you for clarifying that, now I can't believe I didn't see that before! –  Jonn_Underwood Jun 19 '13 at 13:37
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Log is the logarithmic function. ln is the logarithm to the base $e$, which is the Euler number.

Log(x) usually denotes the logarithm of x to base 10.

Ln(x) denotes the logarithm of x to base $e$.

The brackets are often left away to improve readability.

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Neither. Log is a function, the proper notation is really $\ln(x)$ or $\log(x)$, and it is read "log of $x$" or "logarithm of $x$."

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