Page 3 - Calculus for the Clueless, Calc II

4 ln a+ 5 ln b- 6 ln c- ½ ln d

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8. log b b = 1 since b = b. log 7 7 = 1. In e = 1. log 10 = 1.
9. log b 1 = 0 since b = 1. log 8 1 = log 1 = ln1 = 0.
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10. Log is a 1:1 function. This means if log c = log d, c = d.

Note

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Not everything is 1:1. If x = y2, x = ±y.
11. Log is an increasing function. If m < n, then log m < log n.

12.

13. is a weird way of writing x. e = x.
ln x
14. log b b =x; ln e = x.
x
x

15.

You should now be able to solve the following kinds of log equations:

Example 2—
Solve for x: 4 · 3 x + 2 = 28.

Divide by 4; isolate exponent.

Take logs. It now becomes an elementary algebra equation, which we solve for x, using the
same technique as in the implicit differentiation section of Calc 1.

x
16. a = e x ln a . Also x = e x ln x and x sin x = e sin x ln x .
x
Example 3—

Using the same algebraic tricks, we get

Eliminate excess minus signs.

All this should be known about logs before the calculus. Now we are ready to get serious.