Logarithm and exponents questions...

Ok, for the first question:

3^x = 15,
so you know this is an exponential equation, and to balance the equation, you must know logs

so... log3 ( 3^x ) = log3 (15), and the 3 is the base
You have to know log theories as well, so...

logx ( y ), NOT relating to the question.....
= log(y) / log(x)
So...

x = log(15) / log(3)
Hopefully that makes sense, you can also do:
log(3^x) = log(15)
x log(3) = log(15)
x = log(15) / log(3), which is the better more known way to do it, hopefully that makes sense...

For the next one,
log4 (3) = x, so now we know that trick, knowing that the trick can work for any base, not just 10,
log2 (24)
= log4 (24) / log4( 2 )
= log4 (8 * 3) / 1/2
= 2( log4(2*4) + log4 ( 3) )
= 2( log4 (2) + log4( 4) + x )
= 2( 1/2 + 1 + x)
= 3 + 2x

3^x=15
taking log both sides,
log3^x=log15
xlog3=log15
x=log15/log3 (It's a property of log.
log m^n=n logm)

Now the second one:
logbase4(3)=x
=>4^x=3
=>2^2x=3
Note it. It will help us later.

now it's given
logbase2(24)=?
let it's value be n.
=>logbase2(24)=n
=>2^n=24
=>2^n=3*8
=>2^n=2^2x * 2^3 [Since 2^2x=3]
=>2^n=2^2x 3
=>n=2x 3

Log 3^x=log 15
X log 3 = log 15
X= log 15/log 3


Log 3/ log 4=x ---> log 3/2 log 2=x---> log 3/log2=2x
Log 24/log 2=( log 8 + log 3)/ log 2=(3log 2+log3)/log 2=3+2x