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5.1 notes on Logarithmic Functions
The inverse of the exponential function is the logarithmic function. We write this as:
The inverse of the natural exponential function is the natural logarithmic function. We write this as:
Relationships between
logb x and b x
Since exponential functions and logarithmic functions are inverses of each other, for b > 0 and b  1, we have the
following relationships:
a.
b.
c.
d.
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Graph of:
f ( x)  logb x , with b>1
f ( x)  logb x , with 0<b<1
Algebraic Rules for Logarithmic Functions:
For all values of x, y, b, and a for which these expressions are defined, we have
a.
e.
b.
f.
c.
g.
d.
h.
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Examples:
Determine which of the given functions are exponential. Write each exponential function in the form
f ( x)  Ab x
1.
f ( x)  4(1.2) x  5
2.
f ( x)  3.1x 7.4
3.
f ( x)  2 x31 x
Use the fact that y  e x and y  ln x are inverses to simplify the following expressions.
1.
eln 3
2.
e 3ln x
3.
ln(e3 x )
Write the function f ( x)  2 x in the form f ( x)  ekx , and write the function g ( x)  e2 x in the form
g ( x)  b x .
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Calculate each of the following by hand, without a calculator.
1.
log 2 64
2.
log6 3  log 6 12
3.
2log3 6  log3 4
Solve each of the following equations.
1.
3.25(1.72) x  1000
2.
3x  9 x2
Challenge Problem:
9(9x )  10(3x )  1  0
Homework: Properties of Logarithms Worksheet #1-23odd
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