Question

Find the value of x if $ {\left( {150} \right)^{\text{x}}} = 7 $
 $ \begin{align}
  &A.\;\dfrac{{\log 7}}{{\log 3 + \log 5 + 1}} \\
  &B.\;\dfrac{{\log 7}}{{\log 3 + \log 6}} \\
  &C.\;\dfrac{{\log 7}}{{\log 3 + \log 5 + 10}} \\
  &D.\;\dfrac{{\log 7}}{{\log 2 + \log 3}} \\
\end{align} $

Answer 1

Hint:The various concepts and formulas related to logarithms will be used in this question. We can see that x is an exponent, so we will start by taking the logarithm of base 10 on both the sides. Then we will apply the formulas for logarithm that-
 $ \begin{align}
  &\log {a^x} = x\log a \\
  &\log \left( {ab} \right) = \log a + \log b \\
  &\log 10 = 1 \\
\end{align} $

Complete step-by-step answer:
We have to find the value of x in the equation $ {\left( {150} \right)^{\text{x}}} = 7 $ . So, we will first take logarithm of base 10 on both the sides which is-
 $ \begin{align}
  &{\left( {150} \right)^{\text{x}}} = 7 \\
  &Taking\;\log \;on\;both\;sides, \\
  &\log {\left( {150} \right)^{\text{x}}} = \log 7 \\
  &Using\;the\;property\;log{a^{\text{x}}} = xloga, \\
  &x\log\left( {150} \right) = \log 7 \\
\end{align} $
Now we will divide both the sides by log(150), which will bring the required value of x on one side of the equation as-
 $ \begin{align}
  &{\text{x}} = \dfrac{{\log 7}}{{\log 150}} \\
  &We\;know\;that\;150 = 3 \times 5 \times 10 \\
  &Using\;logab = loga + logb, \\
  &\log 150 = \log \left( {3 \times 5 \times 10} \right) = \log 3 + \log 5 + \log 10 \\
  &{\text{x}} = \dfrac{{\log 7}}{{\log 3 + \log 5 + \log 10}} \\
  &We\;also\;know\;that\;\log 10 = 1,\; \\
  &{\text{x}} = \dfrac{{\log 7}}{{\log 3 + \log 5 + 1}} \\
\end{align} $

This is the required value of x. Hence, the correct option is A.

Note: In such types of questions, it is important to take the correct base for the logarithm that we are taking on both the sides. Here, we took a base of 10 because of the requirement of the options. Also, we need to factorize 150 in such a way that it satisfies the option, because there can be various ways to factorize any number.