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Question: Find the value of x if \( {\left( {150} \right)^{\text{x}}} = 7 \) \( \begin{aligned} &A.;\;...

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

Explanation

Solution

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-
log⁡ax=xlog⁡a log⁡(ab)=log⁡a+log⁡b log⁡10=1 \begin{aligned} &\log {a^x} = x\log a \\\ &\log \left( {ab} \right) = \log a + \log b \\\ &\log 10 = 1 \\\ \end{aligned}

Complete step-by-step answer:
We have to find the value of x in the equation (150)x=7{\left( {150} \right)^{\text{x}}} = 7 . So, we will first take logarithm of base 10 on both the sides which is-
(150)x=7 Taking;  log⁡  on  both  sides, log⁡(150)x=log⁡7 Using;  the  property  logax=xloga, x;log⁡(150)=log⁡7 \begin{aligned} &{\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{aligned}
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-
x=log⁡7log⁡150 We;  know  that  150=3×5×10 Using;  logab=loga+logb, log⁡150=log⁡(3×5×10)=log⁡3+log⁡5+log⁡10 x=log⁡7log⁡3+log⁡5+log⁡10 We;  also  know  that  log⁡10=1,   x=log⁡7log⁡3+log⁡5+1 \begin{aligned} &{\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{aligned}

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.