SOLUTION: 90) 3^2x -8 . 3^x +15=0 94) If x= (log125 5)^log5 125 Please help me to solve this two. Thank You

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 90) 3^2x -8 . 3^x +15=0 94) If x= (log125 5)^log5 125 Please help me to solve this two. Thank You       Log On


   

Question 170138This question is from textbook intermediate algebra
: 90) 3^2x -8 . 3^x +15=0
94) If x= (log125 5)^log5 125
Please help me to solve this two. Thank You
This question is from textbook intermediate algebra

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


Table of Contents:
Problem #90
Problem #94




Jump to Top

90)

3%5E%282x%29-8%2A3%5E%28x%29%2B15=0 Start with the given equation


Let z=3%5E%28x%29. So this means that z%5E2=%283%5E%28x%29%29%5E2=3%5E%282x%29 (ie z=3%5E%282x%29)


z%5E2-8z%2B15=0 Replace 3%5E%282x%29 with z%5E2 and 3%5E%28x%29 with "z" (see above)


Notice we have a quadratic equation in the form of az%5E2%2Bbz%2Bc where a=1, b=-8, and c=15


Let's use the quadratic formula to solve for z


z+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


z+=+%28-%28-8%29+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%2815%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-8, and c=15


z+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%281%29%2815%29+%29%29%2F%282%281%29%29 Negate -8 to get 8.


z+=+%288+%2B-+sqrt%28+64-4%281%29%2815%29+%29%29%2F%282%281%29%29 Square -8 to get 64.


z+=+%288+%2B-+sqrt%28+64-60+%29%29%2F%282%281%29%29 Multiply 4%281%29%2815%29 to get 60


z+=+%288+%2B-+sqrt%28+4+%29%29%2F%282%281%29%29 Subtract 60 from 64 to get 4


z+=+%288+%2B-+sqrt%28+4+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


z+=+%288+%2B-+2%29%2F%282%29 Take the square root of 4 to get 2.


z+=+%288+%2B+2%29%2F%282%29 or z+=+%288+-+2%29%2F%282%29 Break up the expression.


z+=+%2810%29%2F%282%29 or z+=++%286%29%2F%282%29 Combine like terms.


z+=+5 or z+=+3 Simplify.


Now remember, we let z=3%5E%28x%29. So this means that 3%5E%28x%29+=+5 or 3%5E%28x%29+=+3


3%5E%28x%29+=+5 Start with the first equation


x+=+log%283%2C%285%29%29 Take the log base 3 of both sides (to isolate "x")

------------------------------------------------

3%5E%28x%29+=+3 Move onto the second equation


x+=+log%283%2C%283%29%29 Take the log base 3 of both sides (to isolate "x")


x+=+1 Evaluate the log base 3 of 3 to get 1


============================================================

Answer:


So the solutions are x+=+log%283%2C%285%29%29 or x+=1








Jump to Top

94)


Is the equation x=%28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29????


If so, then....


x=%28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29 Start with the given equation


Let w=log%28125%2C%285%29%29 and z=log%285%2C%28125%29%29


So the equation then becomes...


x=%28w%29%5E%28z%29


Now let's evaluate the individual pieces:


w=log%28125%2C%285%29%29 Start with the first assignment


w=log%2810%2C%285%29%29%2Flog%2810%2C%28125%29%29 Use the change of base formula to break up the log


w=log%2810%2C%285%29%29%2Flog%2810%2C%285%5E3%29%29 Rewrite 125 as 5%5E3. Note: 5%5E3=125


w=log%2810%2C%285%29%29%2F%283%2Alog%2810%2C%285%29%29%29 Pull the exponent down and place it out front.


w=1%2F3 Divide and cancel out the common terms.


Since w=1%2F3, this means that log%28125%2C%285%29%29=1%2F3 (this is the first piece)


-------------------------------------------------------


z=log%285%2C%28125%29%29 Now move onto the second assignment


z=log%2810%2C%28125%29%29%2Flog%2810%2C%285%29%29 Use the change of base formula to break up the log


z=log%2810%2C%285%5E3%29%29%2Flog%2810%2C%285%29%29 Rewrite 125 as 5%5E3. Note: 5%5E3=125


z=%283%2Alog%2810%2C%285%29%29%29%2Flog%2810%2C%285%29%29 Pull down the exponent.


z=3 Divide and cancel out the common terms.


Since z=3, this means that log%285%2C%28125%29%29=3 (which is the second piece)



x=w%5Ez Go back to the original equation


x=%281%2F3%29%5E3 Plug in w=1%2F3 and z=3


x=%281%5E3%29%2F%283%5E3%29 Distribute the exponent.


x=1%2F27 Cube 1 to get 1. Cube 3 to get 27.



=========================================================

Answer:

So the answer is x=1%2F27


This means that %28log%28125%2C%285%29%29%29%5E%28log%285%2C%28125%29%29%29=1%2F27