What is (log3 x)^2=log3 (x^2) + 3 ?
- print Print
- list Cite
Expert Answers
calendarEducator since 2008
write3,662 answers
starTop subjects are Math, Science, and Social Sciences
(log3 x)^2=log3 (x^2) + 3
First we know that log a^b = b*log a
==> (log3 x)62 = 2log3 x + 3
Now, we will move all terms to the left side so the right side is 0:
==>( log3 x)^2 - 2log3 x - 3 = 0
Now let us assume that:
log3 x = y
==? y^2 - 2y - 3 = 0
Now we will factor:
( y-3)(y+1) = 0
==> y1= 3 ==> log3 x = 3 ==> x = 3^3 = 27
==> y2= -1 ==> log3 x = -1 ==> x = 3^-1 = 1/3
Then the answer is:
x = { 1/3 , 27}
Related Questions
- mathWhat is (log3 x)^2=log3 (x^2) + 3 ?
- 1 Educator Answer
- Write 2 log3 x + log3 5 as a single logarithmic expression.
- 2 Educator Answers
- log3(8x+3) = 1+ log3 (x^2)
- 1 Educator Answer
- Solve for x log2 (x) +log3 (x)=1.
- 1 Educator Answer
- Find x if x^(1+log3 x)=9x^2 .
- 2 Educator Answers
To solve (log 3 x)^2 = log3 (x^2) +3
We know that log 3 (x^2) = 2log3 x by law of logarithms loa^m = m loga.
Therefore the given equation becomes:
(log3 x)^2 = 2log3 x +3.
t^2 = 2t +3 , where t = log3 x.
t^2-2t-3 = 0.
(t+1)(t-3) = 0.
t+1 = 0 , or t-3 = 0.
t = -1 a or t = 3.
t= 3 gives log 3 x= 3. Or x = 3^3 =27.
t =-1 does give any real solution.
Therefore x = 27 is the solution.
Since we have the term log3 x^2, we'll use the power property of logarithms:log3 x^2 = 2 log3 x.
The given equation (log3 x)^2 = log3 x^2 + 3 converts to
(log3 x)^2 - 2log3 x - 3 = 0
We'll substitute log3 x = u.
We'll re-write the equation:
u^2 - 2u -3 =0
Factoring, we'll get:
(u-3)(u+1) =0
We'll put each factor as zero:
u-3 = 0
So, u = 3
u + 1 = 0
u = -1
But log3 x = 3 => x = 3^3 => x = 27
log3 x = -1 => x = 3^-1 => x = 1/3
Since both solutions are positive, we'll accept them.
Student Answers