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How do you simplify \[\log \left( {\dfrac{1}{{{{10}^x}}}} \right)\]?

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Last updated date: 20th Jul 2024
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Answer
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Hint: Here the given question is to simplify for the “log” function, here we know that to simplify this function we need to use standard properties and formula of the “log” function. We know that fraction given in log splits in subtraction and power in the log function comes out as the coefficient of the log function.
Formulae Used: Common log properties is given as:
\[ \Rightarrow \log \left( {\dfrac{a}{b}} \right) = \log (a) - \log (b) \\
   \Rightarrow \log ({x^a}) = a[\log (x)] \\
   \Rightarrow \log 1 = 0 \\
   \Rightarrow \log 10 = 1 \]

Complete step-by-step solution:
Using the standard formulae for log function here in the given question we need to simplify the given term, here first we need to break the fraction into subtraction by using the log property and then by putting the values of the function we can reach toward the simplified answer, on solving here we get:
\[ \Rightarrow \log \left( {\dfrac{1}{{{{10}^x}}}} \right) = \log (1) - \log ({10^x}) = 0 - x\log (10) = - x\]
Here we know:
\[\Rightarrow \log 1 = 0 \\
   \Rightarrow \log 10 = 1 \]
Here we get the simplified value for the given term in the form of log function.

Note: Log function is also a standard function in mathematics like other functions, here to solve for the log function we need to know the associated formulae and the procedure to solve for the value for the function given in the question.