Question

The value of log 0.001 to the base 0.1 is

Answer 1

$\textbf{Given:}$

$\log_{0.1}0.001$

$\text{This can be written as}$

$=\log_{0.1}(0.1)^3$

$\text{Using quotient rule of logarithm}$

$\boxed{\bf\,log_aM^n=n\;log_aM}$

$=3\,\log_{0.1}0.1$

$\text{We know that, }\;\;\boxed{\bf\log_aa=1}$

$=3(1)$

$=3$

$\therefore\textbf{The value of \bf\log_{0.1}0.001 is 3}$

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Answer 2

We have:

$\log_{0.1} 0.001$

We have to find, the value of $\log_{0.1} 0.001$ = ?

Solution:

∴ $\log_{0.1} 0.001$

∵ 0.001 = 0.1 × 0.1 × 0.1 = $0.1^3$

= $\log_{0.1} 0.1^3$

Using the logarithm identity:

$\log m^n$ = n$\log m$

= 3$\log_{0.1} 0.1$

Using the logarithm identity:

$\log_a a$ = 1

= 3 × 1

= 3

∴ $\log_{0.1} 0.001$ = 3

Thus, the value of $\log_{0.1} 0.001$ is equal to 3.