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Precalculus

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Exponentials & Logarithms

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Precalculus: Exponentials and logarithms



1. Definitions: logarithms and exponentials properties

if y = ax , then x = loga(y)
The real positive a is called the base of the logarithm.

If a = 1, then y = 1x = 1 in ]-∞, + ∞[
If a < 0, then y = ax will have no sense when x is rational (for example (-6)1/2)

Conclusion a must be real positive.


2. 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)

Other properties:

loga (a) = 1

loga (1) = 0

loga (ax) = x

a loga(x) = x

loga (xr) = r loga (x)


3. Examples

y = 2x , then x = log2 (y)

y = 10x , then x = log10 (y) = log (x)
called common logarithm.

y = ex , then x = loge y = ln (y)
called natural logarithm.

ln (2) = 0.69

log (10) = 1

log (1/2) = - log 2

4. Exercises

Simplify the following expressions:

ln(a2 b3 c-7)

ln(4 a2/b3/4)

log[4 a3/(b2- 1)]


Solutions



5. Graph of ln(x), log(x) and exp{x}





gnuplot> set border
gnuplot> set xtics 1
gnuplot> set grid
gnuplot> set xzeroaxis lt 2 lw 2
gnuplot> set yzeroaxis lt 2 lw 2
gnuplot> set style line 1 lw 3
gnuplot> set xrange [-7:7]
gnuplot> set yrange [-5:5]
gnuplot> f(x)= log(x); g(x)= log10(x); h(x) = exp (x);
gnuplot> set title "Graph of ln(x), log(x), and exp{x}"
gnuplot> set xlabel " x "
gnuplot> set ylabel "ln(x) and exp{x}"
gnuplot> plot f(x), g(x), h(x) lw 3









  
 



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