Question

How do you solve $ {{\log }_{10}}x=-1 $ ?

Answer 1

Hint: In this question, we are given the value of a logarithmic function and we need to find the value of x. For this we will use the property of logarithm that if $ {{\log }_{b}}a=y $ then $ a={{b}^{y}} $ . Using this we will be able to find the value of x. We will then simplify the value of x using the property of exponent according to which $ {{a}^{-1}}=\dfrac{1}{a} $ .

Complete step by step answer:
Here we are given the logarithmic function as $ {{\log }_{10}}x=-1 $ . We need to find the value of x. Here 10 is the base of log function and the value of log10 when the base is 10 is given as -1. We need to find the value of x for which $ {{\log }_{10}} $ will be equal to -1.
For this we can use a property of logarithm that if we are given that $ {{\log }_{b}}a=y $ . Then we can say that $ a={{b}^{y}} $ i.e. logarithm can be changed to exponential function by certain rules. Here, we are given that $ {{\log }_{10}}x=-1 $ . Comparing with above form we have b = 10, a = x and y = -1.
So \[{{\log }_{10}}x=-1\Rightarrow x={{10}^{-1}}\].
Therefore value of x can be written as \[{{10}^{-1}}\].
Let us now simplify the value of x.
According to the law of exponents we know that $ {{a}^{-1}} $ can be written as $ \dfrac{1}{a} $ . So here we have a = 10. So we can write \[{{10}^{-1}}\] as $ \dfrac{1}{10} $ . So value of x becomes $ \dfrac{1}{10} $ .
Now let us change into decimal. We know, $ \dfrac{a}{10} $ can be written as 0.a so, we have a as 1. So $ \dfrac{1}{10} $ will be written as 0.1
Hence we have the value of x as 0.1.
So x = 0.1.

Note:
 Students should take care of notation and apply the property accordingly. They can use the following method too:
We are given $ {{\log }_{10}}x=-1 $ .
Let us take 10 to the power of each side of the equation we get $ {{10}^{{{\log }_{10}}x}}={{10}^{-1}} $ .
By law of logarithm we know that $ {{a}^{{{\log }_{a}}x}}=x $ so, here $ {{10}^{{{\log }_{10}}x}}=x $ . Hence we get $ x={{10}^{-1}}=\dfrac{1}{10}=0.1 $ which is the same answer.