Algebra & Trig, Sullivan & Sullivan, Fourth Edition
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§4.6 – Exponential, Logarithmic Equations
How do you solve equations that have exponential and/or logarithmic functions in them?
1
3
Ex: 8 x 
or log 2 x  1  4
2
0. Remember that we're solving for x, so our answer will look like x =
1. Use the rules of math we've looked at previously (radicals, quadratic stuff, etc, etc)
2. If it's useful, change from exponential form to logarithmic form, or vice versa
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3
3
Example: Sovle log 2 x  1  4  (rewrite w/o log) 2 4  x  1  16   x  1  3 16 1  x
3. <If there's a log in the equation> Use one of the three rules (product, quotient, power) that
we saw in §13.4.
4. Use one of the rules we'll learn today.
Properties Useful for Solving Exponential / Logarithmic Equations
1. If
2.
x y
then
x
y
a

a
If
ax  ay
then
x y
Example:
5 x  125  5 x  5 3 x = 3
3. If
x y
Example:
4. If
then
log b x  log b y
(x > 0, y > 0)
5 n  20  log 10 5 n  log 10 20  n log 10 5  log 10 20  n 
log b x  log b y
x y
then
(x > 0, y > 0)
Example: Solve log 3 x  2  log 3 7  x  2  7  x  5
Summary from Previous section:
Product Rule For Logarithms:
Example Of Expanding:
log a xy  log a x  log a y
log 7 5 x  log 7 5  log 7 x
Quotient Rule For Logarithms:
Example Of Expanding:
log 10 20
log 10 5
Algebra & Trig, Sullivan & Sullivan, Fourth Edition
log a
x
 log a x  log a y
y
Power Rule For Logarithms:
log a x  N  log a x 
N
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log 3
5
 log 3 5  log 3 7
7
Example Of Expanding:
log 3 x 4  4  log 3 x 
Kind of like omitting the index with radicals means "square root", omitting the base when
working with logs means "log base 10", or the common log.
Other than that, all the previous rules for logs apply.
ICE: Get some practice with common logs
Logarithms – intuitive view
Point out what the graph of a log looks like
 log 0 isn't defined
(0 isn't in the domain of the function – 10 raised to what power is 0?)
 log x , x < 0  a negative number.
As the fraction gets smaller, the number gets more negative
(Plug some of these into a calculator & see – log(.5), log (.05), log (.005)
 log 1  0 ( 10 0  1 )
log y (for 1  y  10 )  between 0 and 1
 log 10  1
log y (for 10  y  100 )  between 1 and 2
 log 100  2
log y (for 100  y  1000 )  between 2 and 3
 log 1000  2
etc.
(Draw this out)
Thus, the log function 'squashes down' stuff – each factor (of 10, in this case) adds one to the
log's result
Example: Richter scale
Summary of Rules For Logs:
Translate from exponential to log form, or vice-versa
y  ax

log a x  y
Product Rule For Logarithms:
log a xy  log a x  log a y
ax  ay
x y

x y
 log b
x  log b y
(x > 0, y > 0)
Algebra & Trig, Sullivan & Sullivan, Fourth Edition
Quotient Rule For Logarithms:
x
log a  log a x  log a y
y
Power Rule For Logarithms:
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ln e x  x
e ln x  x
(for x > 0)
log a x N  N  log a x 
Common Errors:
log a xy  log a x log a y 
log a x  y   log a x  log a y
 x  log a x
log a   
 y  log a y
log a x  y   log a x  log a y