"...I was instructed to round to the closest one ."

You followed that but they gave you F because they did not follow that.

That is your problem, or, that is the problem.

Forgetting that one above...them not following rules..., their answers are following the correct procedure.

They are interpolating, or using proportion, to get to the correct answer. They are not rounding off.

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A = (antilog .7817)(10^3)

According to their table, log 6.05 is .7818 and log 6.04 is .7810. 6.05 is closer, so I used that value and multiplied by 10^3 to find A:

A = 6,050

The accumulated amount of money over 19 years will be $6,050

My answer was given an F. I looked at their answer: $6,049. Obviously they had used a different value for antilog .7817.

I figured out that in order to get $6,049 as the answer, antilog .7817 would have to be 6.049. But how would one figure out to use this logarithm?

Your 0.7817 falls between 0.7810 and 0.7818. So your antilog should fall between 6.04 and 6.05

Difference between 0.7810 and 0.7818 is 0.0008

Diffference between 0.7810 and your 0.7817 is 0.0007

Difference between 6.04 and 6.05 is 0.01

So, you interpolate or solve by proportion. Let c = correction,

c/0.0007 = 0.01/0.0008

c = (0.01)*(0.0007/0.0008) = 0.009

That then should be added to the 6.04.

Hence, your 0.7817 should have an antilog of 6.04 +0.009 = 6.049

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"log P = 3.1472

P = (antilog .1472)(10^3)

There is no exact match in the table. 1.40 is .1461 and 1.41 is .1492. 1.40 is closer, so using that, I get the answer:

P = 1400

$1,400 is the real value of the saved money

Their answer is $1,403. In order to get to this answer, antilog .1472 would have to be 1.403...how could I possibly guess that?"

Same thing. Your 0.1472 falls between 0.1461 and 0.1492. So the antilog of your 0.1472 should fall between 1.40 and 1.41

Difference of 0.1461 and 0.1492 is 0.0031

Difference of 0.1461 and 0.1472 is 0.0011

Difference of 1.40 and 1.41 is 0.01

So,

c/0.0011 = 0.01/0.0031

c = (0.01)(0.0011/0.0031) = 0.00355

Therefore, antilog of 0.1472 = 1.40 +0.0036 = 1.4036