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.