1. Pre-Calc Lesson 5-5
Logarithms
Logarithmic functions--- are the inverse of the
exponential function
Basic exponential function: f(x) = bx
Basic logarithmic function:
f-1(x) = logbx
Every (x,y)  (y,x)

2. Basic rule for changing exponential equations to
logarithmic equations (or vice-versa):
logbx = a  ba = x
The base of the logarithmic form becomes the base
Of the exponential form.
The ‘answer’ to the log statement becomes the power
in the exponential form .
The number you are to take the ‘log’ of in the log
form, becomes the answer in the exponential form.

3. Examples:
log525 = 2 because 52 = 25
log5125 = 3 because 53 = 125
log2(1/8) = - 3 because 2-3 = 1/8
base ‘b’ exponential function
f(x) = bx
Domain: All reals
Range: All positive reals
base ‘b’ logarithmic function
f-1(x) = logb(x)
Domain: All positive reals
Range: All reals

4. There are two special logarithms that your calculator
Types of Logarithms
There are two special logarithms that your calculator
Is programmed for:
log10(x)  called your ‘common logarithm’
For the common logarithm we do not include the
subscript 10, so all you will see is: log (x)
So log10(x)  log (x) = ‘k’ iff 10k = x
loge(x)  called your ‘natural logarithm’
For the natural logarithm we do not include the
subscript e, so all you will see is: ln (x)
So loge(x)  ln (x) = ‘k’ iff ek = x

5. log 6.3 = 0.8 because = 6.3
ln 5 = 1.6 because e1.6 = 5
Example 2 : Find the value of ‘x’ to the nearest
hundredth.
10x = 75
this transfers to the log statement log = x
calculator will tell you  x = 1.88
ex = 75
this transfers to the log statement ln 75 = x
calculator will tell you  x = 4.32

6. Examples:
Evaluate:
a. log 8 2
b. ln 1 e3
c. log
10,000
d. log 5 1
Solve:
a. log x = 4
b. ln x = 1 2
c. log x = - 1.2