If you are not already familiar with the law of tangents, here's an article that can help you:
We will be using the same notations in the following solution.
In the given case, sides a,b are known, and the included angle C. So the sum of angles
A+B=180-C = 180 - 32-28-30 = 147-31-30 = 1967π/2400 radians
By the law of tangents,
In log (to base 10), cannot calculate the values of negative numbers, so we will keep track of the sign ourselves:
log(-tan(A+B)) = -0.1962309
log(-(a-b)) = 1.394801777162711
log(a+b) = 1.830845192308612
log(tan(A+B)*(a-b)/(a+b) = -0.1962309+1.3948018-1.8308452
Antilog(-0.6322743)=0.2331985 as before.
Convert 0.2331985 to degrees,
A = (160-53-11)/2=80-26-36
B = (134-09-49)/2=67-04-55
side c can be found by the cosine rule or the sine rule.
Check my work.
Anon & Drws, thank you for pointing out there is a problem with this solution.
The tangent rule formula that I used was not correct. The formula used for the tangent rule is simply:
(a-b)/(a+b) = tan((A-B)/2)/tan((A+B)/2)
Using the previous values,
Now apply the tangent rule:
(A-B)/2 = atan(1.2580863)= -51.520285
A-B = -103.04057
c can be found by the sine rule or the cosine rule, since 5 of the six unknowns have been calculated. I get c=30.44083 using the cosine rule and it checks with the sine rule.
I believe you can now handle the logarithm part. If you need further help, just post.
I apologize again for the unforgivable mistake in applying the tangent rule formula.