Help with Logarithms- college problems?

Question

The Question asks to use logs to perform the indicated operations:

1. log3 119=?
I put 3^x=119 so x would = 4.35?

2. (.9823)^10

3. FIND x IN Log5 x-Log5 3=2

Thank You so much! I also have to use log to find
87.64/108.2?

and 4 (square root) 17.22?

Answer 1

1) yes.

2) log .9823^10 = 10 log .9823 = 10 (-0.007756) = -0.07756

3) log5 x - log5 3 = log5 (x/3) = 2, so x/3 = 5^2, x=73

Answer 2

Remember that when a^x=b then log(to the base a) of both sides then
log(base a)[a^x] = log(base a)[b}
or x = log(base a)[b]

Indeed, what you wrote, 3^x=119
Try taking log(base10) of both sides; you can look it up in the tables or use a calc.

2. Let x=0.9823^10
Take log(base10) both sides:
log(base10)x =log(base10).9823^10 = 10log(base10)[.9823] but
.9823 = 9.823x10^(-1)so
log(base10)[.9823]=log(10)[9.823x10(-1)]
=log(10)[9.823] - 1 = log(10)[x]
Solve this, then take the antilog

3. log(5)x-log(5)[3] = 2
log(5)x = 2-log(5)[3] now raise both sides to their respective powers
5^log(5)[x] = 5^{2-log(5)[3]} = (5^2){5^[-log(5)3]} = x = 25(-3)