Precalculus: Exponentials and logarithms
1. Recall the following properties
loga (x y ) = loga (x) + loga (y)
loga (x/y) = loga (x) - loga(y)
loga ax = x
To change the base from one to another:
loga (x) = logb (x) . loga (b)
loga (a) = 1
loga (1) = 0
loga (ax) = x
a loga(x) = x
loga (xr) = r loga (x)
2. Exercises
a) Solve for x each of the following equations:
4 e2x + 1 = 1
3 10x - 2 = 30
24 + 7 e2x + 1 = 31
e2x + 1 = - 4
b) Solve for x each of the following equations:
2 log3 (2 + 2 x) = 1
3 + 4 ln( x - 6) = 5
3 + 4 log( x + 6) = - 1
ln(x2 + ln (2 x + 2)) = 0
6 ln(x2/3) - 4 ln (x + 2) = - 2
Solutions
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