Irrational solutions should be expressed precisely and as a decimal rounded to three places.log2⁡(3x+2)−log4⁡x=3

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Step 1  Given,  log2⁡(3x+2)=log4⁡x=3  Step 2  Now,  log2⁡(3x+2)−log4⁡x=3  log2⁡(3x+2)−log22⁡x=3  log2⁡(3x+2)−12log2⁡x=3 (∵ logab⁡c=1bloga⁡c)  log2⁡(3x+2)−log2⁡(x1/2)=3 (∵ alogb⁡c=logb⁡ca)  log2⁡(3x+2x1/2)=3 (∵ logb⁡a−logb⁡c=logb⁡(ac))  3x+2x=23 (∵logb⁡a=c⇒ a=bc)  3x+2x=8  (3x+2)=8x  Squaring both sides,  (3x+2)2=8x  9x2+4+15x=64x  9x2−52x+4=0  x=−(−52)± (−52)2−4× 9× 42× 9  x=52±2704−14418  x=52±256018  x=52±161018  x=4(13±410)18  x=2(13±410)18  x=2(13+4109, 2(13−410)9  are the solutions to the given problem in irrational form.  In decimals, we get, x=5.69980, 0.07797  Thus, after converting the irrationals to decimals and rounding them off to nearest three decimal places, we get, 5.700, 0.0780

Express irrational solutions in exact form and as a decimal rounded to three decimal places. \log_{2}(3x+2)-\log_{4}x=3

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