Solve each equation.Express irrational solutions in exact form and as a decimal

Ramsey

Ramsey

Answered question

2021-10-10

Solve each equation.Express irrational solutions in exact form and as a decimal rounded to three decimal places.
log2(x+1)log4x=1
[Hint: Change  log4x  to base 2.]

Answer & Explanation

hesgidiauE

hesgidiauE

Skilled2021-10-11Added 106 answers

Step 1
Given equation will be solved using properties of logarithms. Difference property of logarithms is logaxlogby=logxy.Exponent property of logarithms is logaxy=ylogax
Change of base formula for logarithms is logab=logcblogca. Another property of logarithm functions that will be used is that they are one-to-one functions hence logax=logay implies x=y.
Step 2
Given equation is log2(x+1)log4x=1. Use properties of logarithms given in step 1 to solve this equation.
log2(x+1)log4x=1
log2(x+1)log2xlog24=1
log2(x+1)log2xlog222=1
log2(x+1)12log2x=1
log2(x+1)log2x12=1
log2{(x+1)x12}=log22
x+1x12=2
x+1=2x12
(x+1)2=4x
x2+2x+1=4x
x22x+1=0
(x1)2=1
x=1
Hence, solution to given equation is x=1.

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