à DEFINITION: For x>0 and 0< a≠ 1, y=logax, if and only if x=ay
Every function of the form f(x)=ax, a>0, a≠1 passes the Horizontal Line Test and therefore must have an inverse.—This inverse function is called the logarithmic function with base a.
NOTE: WHEN READING y=logax, READ LOG OF X TO THE BASE A.
A LOGARITHM IS AN EXPONENT
à The function f(x)= logax is called the logarithmic function with base a.
LOGARITHMIC FORM: y=logax
COMMON LOG: base 10
NATURAL LOG: base e
↓
The function defined by f(x)=logex = lnx, x>0 is the natural logarithmic function.
à EVALUATING LOGARTITHMS:
1. log232 2. log327 3. log31 4. log22
à SOLUTIONS:
1. log232=5 because 25= 32.
2. log327=3 because 33= 27.
3. log31=0 because 30= 1.
4. log22=1 because 21=2.
à REWRITING LOGARITHMIC FUNCTIONS WITH DIFFERENT BASES
BASE B BASE 10 BASE e
logax = logbx logax = log10x logax = lnx
logba log10a lna
à CHANGING BASES USING COMMON LOGARITHMS
log430 = log1030 1.47712 2.4534
log104 à 0.60206 à
à CHANGING BASES USING NATURAL LOGARITHMS
log430 = ln30 à 3.40120 à 2.4535
ln4 1.38629