MODERATE EXERCISES-Logarithms

(3) Rewrite so there are no logarithms of products, powers or quotients.

  1. ln (x3 ex)
  2. log (sin x ) / (x+4)

( HINT , SOLUTION )
SOLUTION:
(a) This is a product inside a logarithm, so:
ln (x3 ex) = ln(x3) + ln(ex).

Now each of these logs can be rewritten using other properties of logs:
ln(x3)= 3ln x and ln(ex) = x .
So ln (x3 ex) = 3ln x + x.

(b) This is a quotient inside a logarithm, so:
log(sin x ) / (x+4) = log(sin x ) - log(x+4).

The first term on the right hand side is a product within a log, so we have:
log(sin x ) - log(x+4) = log() + log(sin x) - log(x+4).

But we can write log() = log x1/2, and then log x1/2 = (1/2)log x. So the final answer is:

log(sin x ) / (x+4) = (1/2)log x + log(sin x) - log(x+4)

(Point to ponder: for what values of x is this expression defined?)


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