Question

`log 1 = x color(white)("dd") x=?`

Answer 1

`x=0`

Explanation:

The concept is:

`log_(base)power="index"`

`power=base^"index"`

`log_(base)1=x`

`1=base^"index"`

Anything raised to the power is zero

`"index"=0`

`x=0`

Answer 2

If `log(1)=x` then `x=0`

A lesson about logs given in the explanation part of the page.

Explanation:

log to base 10 is written as `log_10`.

However, this is the standard form so people tend to leave out the 10 in `log_10` and just write `log`

Ok! We have two choices.

Choice 1. Look it up using a calculator
`color(white)("ddddddd")`In the old days it was obtained from a log book

`color(white)("ddddddd")`By calculator I get: `log_10=0`

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Choice 2. Consider the structure of what a log actually is.
`color(white)("ddddddd")`You know about `3^2`.
`color(white)("ddddddd")`Well the 2 bit is an index. (indices or powers).

Logs are looking at 'powers' but in a different way

Another way of writing `log_10(1)=x" is "10^x=1`

So `x` is the 'power' applied to 10 and this gives the answer 1.

Using examples: it is known that `2^0 = 0,color(white)("d")5^0=1, color(white)("d")365^0=1`

So from this `ubrace(10^0=1)` which is why `log_10(1)=0`
`color(white)("dddddddddd.d")darr`
`color(white)("ddddddddd")10^x=1`