Algebra Know-It-ALL

From the formula for the common logarithm of a product, we can rewrite this as

log 10 y= log 10 10 + log 10 x

But log 10 10 = 1. Therefore

log 10 y= 1 + log 10 x

7. If a positive real number decreases by a factor of exactly 100 (it becomes 1/100 as
   great), then its common logarithm decreases by exactly 2.


8. Let x be the original number, and let y be the final number. We are told that y=x/100.
   Taking the common logarithm of each side of this equation gives us

log 10 y= log 10 (x/100)

From the formula for the common logarithm of a product, we can rewrite this as

log 10 y= log 10 x− log 10 100

But log 10 100 = 2. Therefore

log 10 y= (log 10 x)− 2

9. If a positive real number decreases by a factor of 357, then its natural logarithm
   decreases by ln 357 or, approximately, 5.88.


10. Let x be the original number, and let y be the final number. We are told that y=x/357.
    Taking the natural logarithm of each side of this equation gives us

lny= ln (x/357)

From the formula for the natural logarithm of a ratio, we can rewrite this as

lny= ln x− ln 357

Using a calculator and rounding to two decimal places, we get ln 357 ≈ 5.88, so

lny≈ (ln x)− 5.88

Chapter 29 715