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Question: If \(\log x-5 \log 3=-2\) then x equals? a) 1.25 b) 0.81 c) 2.43 d) 0.8 e) Either 0.8 or 1...

If log⁡x−5log⁡3=−2\log x-5 \log 3=-2 then x equals?
a) 1.25
b) 0.81
c) 2.43
d) 0.8
e) Either 0.8 or 1.25

Explanation

Solution

Hint: First convert the term in the right hand side also to logarithm. Now every term of the equation is a logarithm. Apply basic properties of logarithm, try to keep the variable on the left hand side and remaining all terms on the right hand side. Now remove the log and write the equation of x, by this find the value of x. This value of x is the required result.

Complete step-by-step answer:
Given equation in the question is written in form of:
log⁡x−5log⁡3=−2\log x-5\log 3=-2
By basic logarithmic properties we can say that:
a=log⁡10aa=\log {{10}^{a}}
By applying this to term on the right hand side, we get it as:
log⁡x−5log⁡3=log⁡10−2\log x-5\log 3=\log {{10}^{-2}}
By basic properties of logarithm we can say that:
alog⁡b=log⁡baa\log b=\log {{b}^{a}}
By applying this to the middle term, we can say it as:
log⁡x−log⁡35=log⁡10−2\log x-\log {{3}^{5}}=\log {{10}^{-2}}
By basic knowledge of power of 3, we can say that:
35=243{{3}^{5}}=243
By substituting this into our equation, we can write as;
log⁡x−log⁡243=log⁡10−2\log x-\log 243=\log {{10}^{-2}}
By basic knowledge of power of 10, we can say that:
10−2=1100{{10}^{-2}}=\dfrac{1}{100}
By substituting this value into our equation, we get it as:
log⁡x−log⁡243=log⁡1100\log x-\log 243=\log \dfrac{1}{100}
By adding term log243 on both sides, we get it as:
log⁡x−log⁡243+log⁡243=log⁡1100+log⁡243\log x-\log 243+\log 243=\log \dfrac{1}{100}+\log 243
By cancelling the common terms, we can write it as:
log⁡x=log⁡1100+log⁡243\log x=\log \dfrac{1}{100}+\log 243
By basic properties of logarithm, we know the formula:
log⁡a+log⁡b=log⁡ab\log a+\log b=\log ab
By applying this to our equation we get it as:
log⁡x=log⁡243100\log x=\log \dfrac{243}{100}
By comparing, we can say the values of x in form of
x=243100x=\dfrac{243}{100}
By simplifying, we get it as, x = 2.43. Therefore option (c) is correct.

Note: Alternate method is apply difference formula of left side get log⁡x35\log \dfrac{x}{{{3}^{5}}} and compare to get x35=10−2\dfrac{x}{{{3}^{5}}}={{10}^{-2}} . By solving this you will get the same result. Be careful while applying the log on the right side, if you forget that “-“ you will get it as 100 and the answer will be 24300 which is absolutely wrong.