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22 Algebra Tips and Tricks Part VI (Logarithms)

Logarithms

A logarithm is the inverse function to an exponential function. For example, for the exponential function y=2x, if we have an input of x=6, we get an output of y=64, and we write 64=26. The logarithmic function y=log2(x) is the reverse of this. We swap the input and the output, so now x=64 and y=6. We see 6=log2(64).

In calculus, we will mostly use the exponential function ex and its inverse, ln(x). Below are some important formulas:

eln(x)=xln(ex)=xln(x)+ln(y)=ln(xy)ln(x)ln(y)=ln(xy)aln(x)=ln(xa)

Examples:
 

ln(x2)ln(x).

There are two ways to do this one. First, we can bring down the exponent of two down in front ln(x2)=2ln(x). Then can combine the like terms of 2ln(x) and ln(x):
ln(x2)ln(x)=2ln(x)ln(x)=ln(x)
Alternatively, we can rewrite the subtraction as a division, like so:
ln(x2)ln(x)=ln(x2x)=ln(x)
Either way we get the same answer!
 

ln(e3x4)3ln(x).

First, we rewrite the multiplication using addition. Then we can simply from there.
ln(e3x4)3ln(x)=ln(e3)+ln(x4)3ln(x)=3+4ln(x)3ln(x)=3+ln(x)
 

ln(x).

We know x=x1/2, so ln(x)=ln(x1/2)=12ln(x).
 

ln(xyz3)ln(zxy3).

We can rewrite all the products and divisions as addition and subtraction:
ln(xyz3)ln(zxy3)=ln(x)+ln(y)ln(z3)[ln(z)ln(x)ln(y3)]=12ln(x)+ln(y)3ln(z)ln(z)+12ln(x)+3ln(y)=ln(x)+4ln(y)4ln(z).

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