Precalculus
Notes – Section 3.4
Properties of Logarithms
Since logarithmic functions are the inverses of exponential functions, their properties
are very closely related.
Exponential
Logarithmic
Property name
x a  x b  x a b
log  ab   log a  log b
Product property
xa
log a
 log a  log b
Quotient property
x
xb
x 
a n
 x an
b0  1
x
a
a b
1
 a
x
 b
log  a   n log a
Power Property
logb 1  0
Argument = 1
1
log    log x 1   log x
 x
Negative exponentpower property
n
Rewriting Logarithmic expressions as sums or differences or multiples
of logarithms
log  8 xy 4   log 8  log x  log y 4
 log 23  log x  log y 4
 3log 2  log x  4 log y
x2  5
ln
 ln x 2  5  ln x
x
 ln  x  5  2  ln x
2
1
 1 ln  x 2  5   ln x
2
Rewriting logarithms as a single logarithm
5ln x  2 ln  xy   ln x5  ln  xy 
2
 ln x5  ln  x 2 y 2 
 x5 
 x3 
 ln  2 2   ln  2 
x y 
y 
CHANGE OF BASE FORMULA
y  log4 7
log 4 button, but if we change y  log4 7
Suppose we want to evaluate
4y  7
with a calculator. The problem is that the calculator has no
to exponential form we get:
Now if we take the common logarithm of each side, we get:
log10 4 y  log10 7
Next we use the power property and re-write the equation as:
y log10 4  log10 7
Finally, we divide each side by
y
log10 7
.
log10 4
log10 4 , we obtain:
We have now changed from base 4 in the original equation to
base 10 (common log) in the new equation. Now we can use our calculator to evaluate.
Instead of taking the common logarithm of each side, we could have taken the natural logarithm of
each side and obtained
In general,
y
ln 7
.
ln 4
logb x  loga x
loga b
as long as a, b and x are positive numbers and a  1 and b  1.
This is known as the change of base formula.
So,
y  log4 7 can become y 
1.4037.
log10 7
ln 7
or y 
both of which are approximately equal to
ln 4
log10 4
Evaluating Logarithms using the change of base formula
log 3 16 
ln16
 2.524
ln 3
log 6 10 
log10
 1.285
log 6
ln 2
log 1 2 
 1
2
ln 1
2
 
Evaluating and Solving a Logarithmic Equation]
Solve the given equation for a when x  .125 .
.125=1/8
 a 
log    4 x
 20 
 a 
1
log    4  
 20 
8
 a  1
log   
 20  2
Rewrite in exponential form.
1
a
20
a
10 
20
a  20 10  63.246
10 2 
Solve the given equation for x when a  9 .
 a 
log    .36 x
 25 
 9  9
log   
x
 25  25
25
 9 
log    x
9
 25 
x  1.232
Describe the transformations that must occur to change the graph of
f ( x)  ln x
g ( x)  log 1 x .
5
We work backwards from
g ( x)  log 1 x
5
g(x) becomes g ( x) 
g ( x) 
ln x
after using the change of base formula.
1
ln
5
ln x
using the quotient property in the denominator.
ln1  ln 5
Since ln1  0 , g(x) becomes g ( x) 
Rewrite as g ( x)  
ln x
ln x

0  ln 5  ln 5
1
 ln x  so we can now see the transformations:
ln 5
1) Reflect across the x – axis
2) Vertically shrink by a factor of
1
ln 5
into the graph of
Characteristics of Graphs of Logarithmic Functions
f ( x)  logb x, b  1
y
Domain:  0,  
Range:  ,  
Continuous over domain
Increasing on its domain
No odd or even symmetry
Not bounded above or below
No local extremes
No horizontal asymptotes
Vertical asymptote x  0  the y  axis 
End behavior lim
 logb x   
x 
 b,1
x
1, 0 
