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Logarithmic
and
Exponential
Equations
Steps for Solving a
Logarithmic Equation
CHECK! to make sure your
answer is “legal”
Solve for the variable. If x is in
more than one term get x terms on
one side and factor out the x
Re-write the log equation
in exponential form
If the log is in more than one term,
use log properties to condense
log 4 x  log 4 x  3  1
use the first property of
logs to “condense”
under one log
log a MN  log a M  log a N
log 4 xx  3  1
4  xx  3
1
Re-write the log equation
in exponential form
If the log is in more than one term,
use log properties to condense
log 4 x  log 4 x  3  1
4  x  3x
4  xx  3
1
0  x  3x  4
2
2
x  4x 1  0
x  4,
x  1
Solve for the variable. If x is in
more than one term get x terms on
one side and factor out the x
Re-write the log equation
in exponential form
If the log is in more than one term,
use log properties to condense
log 4 x  log 4 x  3  1
x  4,
x  1
log 4  log 4  3  1
Remember that the
domain of logs is
4
4
numbers greater than 0
so we need to make
not “illegal” since first log is
sure that if we put our
of 4 and second is of 1
answers back in for x
we won’t be trying to
4
4
take the log of 0 or a
negative number.
ah oh---can’t take the log of - 1 or - 4
so must throw this solution out
log 1  log
CHECK! to make sure your
answer is “legal”
1  3  1
Steps for Solving an
Exponential Equation
Solve for the variable. If x is in
more than one term get x terms on
one side and factor out the x
Use the 3rd Property of Logs to
move the exponent out in front
If you have au = bv and you can’t express a and b
with the same base, take the log of both sides
(ln or log)
x 1
2 x 3
x 1
5
ln 2  ln 5
x 1ln 2  2x  3ln 5
x ln 2  ln 2  2x ln 5  3 ln 5
2
2 x 3
Solve for the variable. If x is in
more than one term get x terms on
one side and factor out the x
Use the 3rd Property of Logs to
move the exponent out in front
If you have au = bv and you can’t express a and b
with the same base, take the log of both sides
(ln or log)
2
x 1
5
2 x 3
x ln 2  ln 2  2x ln 5  3 ln 5
x ln 2  2x ln 5  ln 2  3 ln 5
x ln 2  2 ln 5  ln 2  3 ln 5
ln 2  2 ln 5 ln 2  2 ln 5
put in calculator
making sure to
enclose the
numerator in
parenthesis and
the denominator in
parenthesis
ln 2  3 ln 5
x
 2.186
ln 2  2 ln 5
Solve
Solvefor
forthe
thevariable.
variable. IfIfxxis
is in
in
more
morethan
thanone
oneterm
term get
getxxterms
terms on
on
one
oneside
side and
andfactor
factor out
outthe
thexx
If you have an exponential equation with a base “e”
then isolate the e meaning get rid of other terms and
coefficients and then take the ln of both sides.
Remember that e’s and ln’s are inverses and “undo”
each other so they’ll cancel out and you can solve from
there.
2 x 1
26
4
2 x 1
e

3
4
2 x 1
ln e
 ln
3
3e
isolate the “e”
3e
2 x 1
take the ln of both sides
4
2 x  1  ln
3
4
ln  1
3
x
 .644
2
4
Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah
USA for her hard work in creating this PowerPoint.
www.slcc.edu
Shawna has kindly given permission for this resource to be downloaded
from www.mathxtc.com and for it to be modified to suit the Western
Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar
www.ststephens.wa.edu.au