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Question Number 1355 by Rasheed Ahmad last updated on 25/Jul/15

8^(log (12x+1)) =4^(log 27)    ,solve for x.

$$\mathrm{8}^{{log}\:\left(\mathrm{12}{x}+\mathrm{1}\right)} =\mathrm{4}^{{log}\:\mathrm{27}} \:\:\:,{solve}\:{for}\:{x}. \\ $$

Answered by Yugi last updated on 25/Jul/15

Rewriting the above equation in base 2 gives                    2^(3log(12x+1)) =2^(2log27) ..........(1)  Since the bases are the same, we can equate  the indices on both sides of  (1).  ∴ 3log(12x+1)=2log3^3   3log(12x+1)=6log3  ÷3: log(12x+1)=log9  Hence 10^(log(12x+1)) =10^(log9)   ∴ 12x+1=9 ⇒x=(2/3)

$${Rewriting}\:{the}\:{above}\:{equation}\:{in}\:{base}\:\mathrm{2}\:{gives} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{3}{log}\left(\mathrm{12}{x}+\mathrm{1}\right)} =\mathrm{2}^{\mathrm{2}{log}\mathrm{27}} ..........\left(\mathrm{1}\right) \\ $$$${Since}\:{the}\:{bases}\:{are}\:{the}\:{same},\:{we}\:{can}\:{equate} \\ $$$${the}\:{indices}\:{on}\:{both}\:{sides}\:{of}\:\:\left(\mathrm{1}\right). \\ $$$$\therefore\:\mathrm{3}{log}\left(\mathrm{12}{x}+\mathrm{1}\right)=\mathrm{2}{log}\mathrm{3}^{\mathrm{3}} \\ $$$$\mathrm{3}{log}\left(\mathrm{12}{x}+\mathrm{1}\right)=\mathrm{6}{log}\mathrm{3} \\ $$$$\boldsymbol{\div}\mathrm{3}:\:{log}\left(\mathrm{12}{x}+\mathrm{1}\right)={log}\mathrm{9} \\ $$$${Hence}\:\mathrm{10}^{{log}\left(\mathrm{12}{x}+\mathrm{1}\right)} =\mathrm{10}^{{log}\mathrm{9}} \\ $$$$\therefore\:\mathrm{12}{x}+\mathrm{1}=\mathrm{9}\:\Rightarrow{x}=\frac{\mathrm{2}}{\mathrm{3}}\: \\ $$

Commented by Rasheed Soomro last updated on 25/Jul/15

Thanks. Appreciation for your approach.

$${Thanks}.\:{Appreciation}\:{for}\:{your}\:{approach}. \\ $$

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