Question

How do you write \[\log 0.001 = x\] in exponential form?

Answer 1

Hint: Here, we will convert the given logarithmic equation in the exponential form by using the logarithmic rule. Then we will rewrite the decimal number as a multiple of 10. Then we will equate the exponent of the terms to get the value of \[x\]. Then again substituting this value in the exponential equation we will get the required answer.

Formula Used:
If \[{\log _{10}}x = m\] ,then \[x = {a^m}\]

Complete Step by Step Solution:
We are given that \[\log 0.001 = x\].
We know that the common logarithm always has a base \[10\]. So, we get
\[ \Rightarrow {\log _{10}}0.001 = x\]
If \[{\log _{10}}x = m\] ,then \[x = {a^m}\].
By using the logarithmic rule, we get
\[ \Rightarrow 0.001 = {10^x}\]………………………………..\[\left( 1 \right)\]
Now, we will change the other such that both the sides of the equation have the same base, so we get
\[ \Rightarrow {10^x} = 1 \times {10^{ - 3}}\]
\[ \Rightarrow {10^x} = {10^{ - 3}}\]
By canceling the bases, we get
\[ \Rightarrow x = - 3\]
Substituting \[x = - 3\] in the equation \[\left( 1 \right)\], we get
\[ \Rightarrow 0.001 = {10^{ - 3}}\]

Therefore, the exponential form of \[\log 0.001 = x\] is \[{10^{ - 3}}\].

Note:
We know that a logarithmic equation is an equation that involves the logarithm of an expression with a variable on either of the sides. An exponential function is defined as a function in a variable written in exponents. The given equation is of a second type such that only one side of the equation has a logarithmic function, then the equation on the right becomes the exponent of the base of the logarithm. i.e., \[{\log _b}M = N \to M = {b^N}\]. We know that the logarithmic and exponential are inverses to each other. The logarithm with base 10 is called a common logarithm. The logarithm with the base \[e\] is called the natural logarithm. The given logarithm equation is a common logarithm with base 10. We can equate two equations only when the functions on either of the sides are equal.