SOLUTION: log base 4 of y + log base2 of y =12

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Question 1205634: log base 4 of y + log base2 of y =12


Found 3 solutions by MathLover1, ikleyn, greenestamps:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

log+%284+%2Cy+%29%2B+log+%282+%2Cy%29+=12.......change to base 10

log+%28y+%29%2Flog+%284%29%2B+log+%28y%29+%2Flog+%282+%29=12

%28log+%28y+%29log+%282%29%2B+log+%28y%29log+%284%29%29+%2F%28log+%284%29log+%282+%29%29=12

log+%28y+%29log+%282%29%2B+log+%28y%29log+%284%29+=12%28log+%284%29log+%282+%29%29

log+%28y+%29%28log+%282%29%2B+log+%284%29+%29=12%28log+%284%29log+%282+%29%29

log+%28y+%29%28log+%282%2A4%29+%29=12%28log+%284%29log+%282+%29%29

log+%28y+%29%2Alog+%282%5E3%29+=12%28log+%284%29log+%282+%29%29

log+%28y+%29%2A3log+%282%29+=12%28log+%284%29log+%282+%29%29

log+%28y+%29=%2812%28log+%284%29log+%282+%29%29%29%2F%283log+%282%29+%29....simplify



log+%28y+%29=4%2Alog+%284%29

log+%28y+%29=log+%284%5E4%29

y=4%5E4

y=256

Answer by ikleyn(52887) About Me  (Show Source):
You can put this solution on YOUR website!
.
log base 4 of y + log base2 of y =12
solve for y
~~~~~~~~~~~~~~~~~~~~ 

Your starting equation is

    log%284%2C%28y%29%29 + log%282%2C%28y%29%29 = 12.    (1)


Use the base change formula

    log%282%2C%28y%29%29 = log%284%2C%28y%29%29%2Flog%284%2C%282%29%29 = log%284%2C%28y%29%29%2F%28%281%2F2%29%29 = 2%2Alog%284%2C%28y%29%29.


Therefore, equation (1) takes the form

    log%284%2C%28y%29%29 + 2%2Alog%284%2C%28y%29%29 = 12

    3%2Alog%284%2C%28y%29%29 = 12

    log%284%2C%28y%29%29 = 12/3 = 4

    y = 4%5E4 = 256.


ANSWER.  y = 256.

Solved.

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

On logarithms and their properties,  see these introductory lessons
    - WHAT IS the logarithm
    - Properties of the logarithm
    - Change of Base Formula for logarithms
    - Evaluate logarithms without using a calculator
    - Simplifying expressions with logarithms
    - Solving logarithmic equations
in this site.



Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


All the steps in the solution from tutor @MathLover1 are valid, and she ends up with the right answer. But the solution is extremely awkward, and quite lengthy.

Basically to solve an equation involving logarithms you need to get the logarithms all with the same base. Tutor @ikleyn shows one way to do that in her response, leading to a quick and easy solution.

When you get more familiar with logarithms, you can quickly and easily convert the given equation to an equation involving only a single base informally, by reasoning like this:

Since 4 is 2 squared, log base 2 of A is 2 * log base 4 of A.

Here is a formal display of this process:

log%284%2C%28y%29%29=A --> 4%5EA=y --> %282%5E2%29%5EA=y --> %282%29%5E%282A%29=y --> log%282%2C%28y%29%29=2A

When you have this level of familiarity with logarithms, the solution of the given equation becomes easier:

log%284%2C%28y%29%29%2Blog%282%2C%28y%29%29=12
log%284%2C%28y%29%29%2B2log%284%2C%28y%29%29=12
3log%284%2C%28y%29%29=12
log%284%2C%28y%29%29=4
y=4%5E4=256