ALG 2: exponents & logarithms?

Question

Please help me with the following problems. Even though given the answers, I need help on how to get that answer. Thank you.

1) Given that log(base2)3 = 1.6, find an approximation to one decimal place of log(base2)144. (Answer: 7.2)

2) Evaluate a ^ [2log (base a)3 + b ^ 3log (base b)2] (Answer: 17)

3) Solve for x: 5 ^ (x) – 5 ^ (x-2) = 120squareroot5 (Answer: 3 ½)

4) log (base squareroot2) times cube root of 2 (Answer: 2/3)

Answer 1

1.) Break 144 down into it's divisors:
144 = 12^2 = 3^2 * 2^4
log (2) 144 = log (2) (3^2 * 2^4)

Logarithmic rule: log ab = log a + log b.
log (2) (3^2 * 2^4) = log (2) 3^2 + log (2) 2^4

Logarithmic rule: log b^a = a log b
log (2) 3^2 + log (2) 2^4 = 2 log (2) 3 + 4 log (2) 2
2 log (2) 3 + 4 log (2) 2 = 2 (1.6) + 4 * 1 = 3.2 + 4 = 7.2

2.) This one's a little too confusing.

4.) This should be written as log (√2) 2(1/3)

You can square the base and the number in the logarithm:
log (10) 100 = log (100) 10000
so
log (2) (2(1/3))^2 = log (2) (2(2/3)) = 2/3