Question

Solve the following equation:
$\log (\log x) + \log (\log {x^4} - 3) = 0$

Answer 1

Hint: In order to solve this question, we will use some properties of logarithmic functions such as sum of the logarithmic terms $\log a + \log b = \log (a \times b)$ and ${\log _b}a = c{\text{ then }}a = {b^c}$ By using sum of the logarithmic term property we will get an quadratic equation and by solving it we will get the answer.

Complete Step-by-Step solution:
Given equation
$\log (\log x) + \log (\log {x^4} - 3) = 0$
By using sum of the logarithmic terms property, we will simplify our equation
$\because \log a + \log b = \log (a \times b)$
$
   \Rightarrow \log \left[ {\log (x)(\log {x^4} - 3)} \right] = 0 \\
   \Rightarrow \log \left[ {\log (x)(\log {x^4} - 3)} \right] = \log 1{\text{ }}\left\{ {\because \log 1 = 0} \right\} \\
$
By simplifying it further, we get
\[ \Rightarrow \log (x)(\log {x^4} - 3) = 1\]
As we know \[\left[ {\log {a^b} = b\log a} \right]\]
$
   \Rightarrow \log x(4\log x - 3) = 1 \\
   \Rightarrow 4{\left( {\log x} \right)^2} - 3\log x - 1 = 0 \\
$
Now, for solving above quadratic equations, we will assume $\log x = t$
So, the equation become as
$
   \Rightarrow 4{t^2} - 3t - 1 = 0 \\
   \Rightarrow 4{t^2} - 4t + t - 1 = 0 \\
   \Rightarrow4t(t - 1) + 1(t - 1) = 0 \\
   \Rightarrow (4t + 1)(t - 1) = 0 \\
   \Rightarrow t = 1{\text{ or }}\dfrac{{ - 1}}{4} \\
$
We have the roots of the quadratic equation as $t = 1{\text{ or }}\dfrac{{ - 1}}{4}$
By taking $t = \dfrac{{ - 1}}{4}$
Put the value of t as log x
$ \Rightarrow {\log _{10}}x = \dfrac{{ - 1}}{4}$
As we know ${\log _b}a = c{\text{ then }}a = {b^c}$
\[
  \therefore x = {(10)^{\dfrac{{ - 1}}{4}}} \\
  x = \dfrac{1}{{\sqrt[4]{{10}}}} \\
\]
By taking $t = 1$
Putting the value of t as $\log x$
$ \Rightarrow {\log _{10}}x = 1$
As we know ${\log _b}a = c{\text{ then }}a = {b^c}$
\[
  \therefore x = {(10)^1} \\
  x = 10 \\
\]
Hence, the solution of given equation is \[x = \dfrac{1}{{\sqrt[4]{{10}}}}{\text{ and }}10\]

Note: In order to solve problems related to logarithms we must be aware of the properties of logarithm and exponents. And also must have a good command on solving algebraic equations. For example in the above question we not only used the concept or properties of logarithm but also relation between exponents and logarithms. Remember the concept of domain and range of these functions, in some problems it is quite useful.