Any Other Way To Solve For 170,000=130*1.03^x Than Using Logarithm?

You could use trial and error, or you could use some (other) form of iteration. If you choose Newton's method, the new value of x (call it x') will be found from the old value by

  x' = x - (1 - 170,000/130*1.03^(-x))/Ln[1.03]

Of course, this involves the logarithm Ln[1.03], which can be reasonably approximated by .03 in this equation. When I started with x=120*, I found it took about 38 iterations to compute x to 6 decimal places. (The final value was 242.771.)

Virtually all scientific calculators, including the one that comes standard with Windows, have logarithm functions available, as do virtually all spreadsheet programs. On the web are numerous scientific calculators, and the Google search box (calculator) is able to compute them, too.

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* I chose 120 due to a mistake in mental math. The "rule of 72" says that the doubling time at an interest rate of n% is 72/n years. While not completely accurate, it can get you in the ballpark. The ratio 170,000/130 is about 1307, which is "close" to 1024, 10 doublings. I erroneously computed 72/3 as 12, so chose x=10*12 as a starting value. That is why so many iterations were required. Had I correctly chosen 240, only 3 or 4 iterations would have been required to get to the proper answer.