Compare the above with the result of entering −3 into a scientific calculator, then hitting
“Inv,” then hitting “ln,” and finally rounding to four decimal places:e−^3 ≈ 0.0498Rounding error stayed out of this little exercise!Product vs. sum
Exponential functions can express the relationship between sums and products, just as loga-
rithms do. Suppose that x and y are real numbers. Thenbxby=b(x+y)whenb > 0. To demonstrate, let b= 10, x= 4, and y=−6. We can plug in the numbers for the
product of exponentials and get and get104 × 10 −^6 = 10,000 × 0.000001
= 0.01When we evaluate the right side, we get10 [4+(−6)]= 10 (4−6)
= 10 −^2
= 0.01The results agree. You’ll find that this is always true, no matter what base and arguments you
use, as long as the base is positive. Of course, if you get nonterminating decimals for any of
the values in the calculation, you should expect some rounding error.Ratio vs. difference
Again, suppose that x and y are real numbers. Thenbx / by=b(x−y)whenb > 0. Using the same numerical values as before, we can demonstrate this. We plug in
the numbers on the left side of the equation and get104 / 10−^6 = 10,000 / 0.000001
= 10,000 × 1,000,000
= 10,000,000,000
= 1010Then we can evaluate the right side to see that10 [4−(−6)]= 10 (4+6)
= 1010492 Logarithms and Exponentials