SOLUTION: 1. Solve for x: {{{ 2log_64(3x-2)=1/3 }}} 2. Write without logarithms and negative exponents: {{{ 2log_3root(3,81)+e^(-4ln(t)) }}} 3. Solve equation: x^(1/2)+3x^(1/4)=10 4

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 1. Solve for x: {{{ 2log_64(3x-2)=1/3 }}} 2. Write without logarithms and negative exponents: {{{ 2log_3root(3,81)+e^(-4ln(t)) }}} 3. Solve equation: x^(1/2)+3x^(1/4)=10 4      Log On


   

Question 969212: 1. Solve for x: +2log_64%283x-2%29=1%2F3+
2. Write without logarithms and negative exponents: +2log_3root%283%2C81%29%2Be%5E%28-4ln%28t%29%29+
3. Solve equation: x^(1/2)+3x^(1/4)=10
4. +Let+H%28t%29+=+sqrt%2820-t%5E2%29+-+2 find domain
Thank you!!!!
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Interesting problems. Here is a way to deal with #3.

Let v=x%5E%281%2F2%29. This means x%5E%281%2F4%29=v%5E2.

v%2B3v%5E2=10
3v%5E2%2Bv-10=0
discriminant is 1%5E2%2B4%2A3%2A10=1%2B120=121=11%5E2.

v=%28-1%2B-+sqrt%2811%5E2%29%29%2F%282%2A3%29
v=%28-1%2B-+11%29%2F6
v=-2 or v=5%2F3

Substitute back.
x%5E%281%2F2%29=-2 or x%5E%281%2F2%29=5%2F3
highlight%28x=4%29 OR highlight%28x=25%2F9%29.



Just a start in attempting #2:

2%2Alog%283%2C%2881%29%5E%281%2F3%29%29%2Be%5E%28-4ln%28t%29%29

2%2Alog%283%2C%28%283%5E4%29%5E%281%2F3%29%29%29%2Be%5E%28-4ln%28t%29%29

2%2Alog%283%2C%283%5E%284%2F3%29%29%29%2Be%5E%28-4ln%28t%29%29

2%284%2F3%29%2Alog%283%2C3%29%2Be%5E%28-4ln%28t%29%29

8%2F3%2Be%5E%28-4%2Aln%28t%29%29

...think what you can do with that last term with e....
...work with the exponent.

8%2F3%2Be%5E%28ln%28t%5E%28-4%29%29%29, you need to recognize the rule of logarithms used in the step.
Now recognize ln() function and e^() function are inverses of each other.

8%2F3%2Bt%5E%28-4%29

highlight%288%2F3%2B1%2Ft%5E4%29