Please help me with these math problems: exponents/logs?

Question

Solve for x
1.12^(2x+3)=6^(2x+3)
2.10^(3x)=5e^(x-2)
3. 4^(2x)=2(5)^(x-3) I forgot how to do these ones if you could help me i would greatly appreciate it

Answer 1

1) Take log/natural log (whichever your more comfortable with) of both sides.

ln(12^(2x+3)) = ln(6^(2x+3))

expononets can be put on the outside, do this and you get...

(2x + 3)(ln 12) = (2x + 3)(ln 6)
2xln12 + 3ln12 = 2xln6 + 3ln6
2xln12 + 2xln6 = 3ln12 + 3ln6
x(2ln12 + 2ln6) = 3ln12 + 3ln6
x = (3ln12 + 3ln6)/(2ln12 + 2ln6)
x = 1.5

You could of also noticed from the begging that the same two numbers to the same exponent, then the exponent has to be zero. (2x + 3 = 0, x =1.5)

2) Natural log both sides:
ln(10^3x) = ln(5e^(x-2))
same as last time on the left, on right use the fact that lnab = lna + lnb

3xln10 = ln5 + lne^(x-2)
The natural log of e to any exponent is the exponent
3xln10 = ln5 + x - 2
3xln10 -x =ln5 -2
x(3ln10 -1) = ln5 -2
x = (ln5 -2)/(3ln10 - 1)
that is the exact answer, it comes to approx. -.06611

3) This is almost identical to the last

2xln4 = ln2 + (x - 3)ln5
2xln4 -xln5 =ln2 - 3ln5
x = (ln2 - 3ln5)/(2ln4 -ln5)
approx -3.555

Answer 2

1. divide both sides by 6^(2x+3), getting 2^(2x+3) = 1, so
2x + 3 = 0
x = -3/2

2. take ln of both sides:
3x ln 10 = ln 5 + x - 2
3x ln 10 - x = ln 5 - 2
(3 ln 10 - 1)x = ln 5 - 2
x = (ln 5 - 2)/(3 ln 10 - 1)

3. 2x ln 4 = ln 2 + x - 3
2x ln 4 - x = ln 2 - 3
(2 ln 4 - 1)x = ln 2 - 3
x = (ln 2 - 3)/(2 ln 4 - 1)