ALGEBRA III
NOTES(6.7)
NAME ________________
PERIOD _______________
CHANGE OF BASE FORMULA
I. The change-of-base formula provides a way to evaluate logarithms of bases
other than 10 or e.
Change-of-base:
logb x 
log x
log b
Notice that the formula changes the original logarithm into the quotient of
common logs. At this point, you can use the calculator to evaluate the common
logs and then simplify.
Example:
log4 30 
log30 1.477

log 4
.602
Evaluate the following:
1)
log2 14 
2)
log3 5 
3)
log4 10 
4)
log9 5 
2.453
II.
Using Logarithms to Solve Exponential Equations:
One way to solve exponential equations is to use the property that if two
powers with the same base are equal, then their exponents must be equal.
If b x  b y , then x  y .
Example:
43x  8x 1
Solve
2 
2 3x
 23 
x 1
26x  23x 3
So
6x  3x  3
3x  3
x 1
However, when each side of an exponential equation cannot easily be expressed
in terms of the same base, logarithms can be used to solve the exponential
equation.
Example:
Solve 4x  7
Change to logarithmic form.
log4 7  x
Rewrite using change-of-base
x 
Use the calculator to evaluate
the common logs. Then simplify.
Example:
Solve
log7
log 4
log7 .8451

 1.4036
log 4 .6021
3x 1  5
Change to logarithmic form.
log3 5  x  1
Rewrite using change-of-base
log5
 x 1
log3
Evaluate the common logs and
solve for x.
.6999
 x 1
.4771
1.4651  x  1
.4651  x
Example:
Solve
32x  4
Change to logarithmic form
log3 4  2x
Rewrite using change-of-base
log 4
 2x
log3
Evaluate the common logs and
solve for x.
.6021
 2x
.4771
1.2620  2x
.6310  x
Try this:
Solve
2x  7