Question

How do you evaluate \[\log 0.0001\]?

Answer 1

Hint: Here we will evaluate the \[log\] term by using the relation between logarithm and exponent. First, we will use the logarithm property to write the value in exponential form. Then we will make the base of both values the same on both sides of the equal sign. Finally, we will compare the power of the term on both sides to get the required answer.

Complete step by step solution:
By the definition of the logarithm:
\[{\log _a}\left( y \right) = x\]
This is also written as,
\[y = {a^x}\]
We have to evaluate \[\log 0.0001\]
So, let us take \[\log 0.0001 = x\]…..\[\left( 1 \right)\]
Now, comparing it with \[{\log _a}\left( y \right) = x\] we get,
\[\begin{array}{l}a = 10\\y = 0.0001\end{array}\]
\[x = x\]
Now using property \[y = {a^x}\] we can rewrite equation \[\left( 1 \right)\] as,
\[{10^x} = 0.0001\]
We can write \[0.0001\] as \[{10^{ - 4}}\] because we have to make the base the same on both sides.
So, \[{10^x} = {10^{ - 4}}\]
On comparing the exponents, we get
\[x = - 4\]

Therefore, the value of \[\log 0.0001 = - 4\].

Note:
A logarithm is a power with which we raise a number to get some other number. It is the inverse function of exponentiation. A logarithm is the opposite of exponential in the same way as subtraction is the opposite of addition. We know that Log sign undo exponentials and therefore another word for exponents is power. Exponential happens when a number is raised to a particular power whereas logarithm is the exponent that a base has to be raised to make that number. If two logarithm terms are added they can be written together in product form under one logarithm sign. However, if two logarithm terms are subtracted we can write them as one term divided by another under the same logarithm sign.