Solve each logarithmic equation.Express irrational solutions in exact form and as a decimal rounded to three decimal places. log(2x+1)=1+log(x-2)

Mark Rosales

Mark Rosales

Answered question

2022-11-04

Solve each logarithmic equation.Express irrational solutions in exact form and as a decimal rounded to three decimal places.
log ( 2 x + 1 ) = 1 + log ( x 2 )

Answer & Explanation

mentest91k99

mentest91k99

Beginner2022-11-05Added 17 answers

Step 1)
The commonly used bases for logarithms are e,10. Base e logarithm is called natural logarithm. If not specified otherwise the logarithm is assumed to be natural logarithm.
One property of natural logarithm that will be used is loge=1. Another property of logarithm that will be used is loga+logb=logab. After solving for x check that the logarithmic functions in given equation are defined for that value.
Given equation is log(2x+1)=1+log(x-2)
Use properties mentioned in Step 1 to solve for x.
log  ( 2 x + 1 ) = 1 + log  ( x  2 )  log  ( 2 x + 1 ) = log  e + log  ( x  2 )  log  ( 2 x + 1 ) = log  e ( x  2 )  2 x + 1 = e ( x  2 )  2 x + 1 = e x  2 e  2 x + 1  e x = e x  2 e  e x  ( 2  e ) x + 1 =  2 e  ( 2  e ) x + 1  1 =  2 e  1  ( 2  e ) x =  2 e  1  x =  2 e  1 2  e  = 1 + 2 e e  2 
This value of x is positive so log  ( 2 x + 1 ) is defined. Since e-2<1 hence this value of x is greater than 2 hence log  ( x  2 ) is defined for this value.
Hence the solution to given equation is 1 + 2 e e  2 

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