Question

If \[{\log _{10}}\left( {{{\log }_{10}}\left( {{{\log }_{10}}x} \right)} \right) = 0\] , then \[x = \]
A) \[1000\]
B) \[{10^{10}}\]
C) \[10\]
D) \[0\]

Answer 1

Hint:
Here, we have to solve for the variable. We will be using the logarithmic rule for converting the logarithm function into an exponential equation to solve the given logarithmic equation. A logarithmic equation is an equation that involves the logarithm of an expression with a variable on either of the sides.

Complete step by step solution:
We are given that \[{\log _{10}}\left( {{{\log }_{10}}\left( {{{\log }_{10}}x} \right)} \right) = 0\]
The logarithmic rule states that if \[{\log _{10}}x = m\], then \[x = {a^m}\] .
By using the logarithmic rule, we get
\[ \Rightarrow {\log _{10}}\left( {{{\log }_{10}}x} \right) = {10^0}\]
When any number is raised to the power zero, then it would be one. So, we get
\[ \Rightarrow {\log _{10}}\left( {{{\log }_{10}}x} \right) = 1\]
Again using the logarithmic rule, we get
\[ \Rightarrow {\log _{10}}x = {10^1}\]
\[ \Rightarrow {\log _{10}}x = 10\]
By using the logarithmic rule, we get
\[ \Rightarrow {\log _{10}}x = 10\]
\[ \Rightarrow x = {10^{10}}\]

Therefore, the value of \[x\] is \[{10^{10}}\].

Additional Information:
We usually isolate the logarithmic term before we convert the logarithmic equation to an exponential function. We will solve all the logarithmic equations by using the properties of logarithms. If both sides of the equation have logarithmic functions then it is easy to solve the variable by cancelling the logarithm on both the sides and then equating the arguments to find the variable.

Note:
The given equation is of 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. The logarithm with base 10 is called common logarithm. The logarithm with the base \[e\] is called natural logarithm. The given logarithm equation is a common logarithm with base 10.