Notes: Solving Logarithmic Equations - wclark.k12.in.us

Notes: Solving Logarithmic Equations
Date _________
• Common Logarithm
1) __A common logarithm is a logarithm that uses base 10____________
ex: log 10 x = log x_______________________________
2) _______________________________________________________________
• If you do not have a common log (base 10) then the Change of Base Formula
must be used in order to enter it in the calculator.
log b
log
a
b
log a
(Note: Round answers to 3 decimal places when necessary.)
ex 1: log 3 81= log 81 / log 3 = 4__________________________
ex 2: log 4 9 = log 9 / log 4 = 1.585________________________
ex 3: log 6 14 = log 14 / log 6 = 1.473_______________________
• Natural Log
1) A natural logarithm is a logarithm that uses base e___________
2) e is an irrational number (approx.) 2.71828 . . . _____________________
ex 4: ln 1 = 0
ex 5: ln 4 = 1.386
__e 0 = 1____________
__e 1.386 = 4____________
(Note: There is a ln button on your calculator along with its inverse e x .)
• If a log (or ln) equation has all numbers in the log, simply use a calculator to
solve.
ex 6: log 4
2
1 = x
ex 7: ln 20 = x
log (1/2) = - 2 x = 2.996
log (4)
ex 8: log 2 (-20) = x None – Can’t have the log of a negative number no
matter what base it is

• If a logarithm equation has a variable in the log, then it must be rewritten in its
exponential form to solve.
• Note: The log must be isolated before being rewritten in its exponential form.
ex 9: log 3 x = 5 ex 10: log x 64 = 3
3 5 = 343 x 3 = 64
[x 3 ] 1/3 = (64) 1/3
x = 4
ex 11: log 2 (x + 1) = 1 ex 12: log 2 x + 1 = 1
2 1 = x + 1 log 2 x + 1 = 1
2 = x + 1 log 2 x = 0
1 = x 2 0 = x
1 = x
ex 13: log (3x + 1) = 5 ex 14: 3 log (2x – 2) = 12
10 5 = 3x + 1 log (2x – 2) = 4
100000 = 3x + 1 10 4 = 2x + 2
99999 = 3x 10000 = 2x+ 2
33333 = x 10002 = 2x
5001 = x
ex 15: log 5x + 3 = 9 ex 16: log (7 – 2x) = -1
log 5x + 3 = 9 log (7 – 2x) = -1
log 5x = 6
10 -1 = 7 - 2x
10 6 = 5x 1/10 = 7 – 2x
1,000,000 = 5x 1/10 – 7 = - 2x
200,000 = x - 6.9 = - 2x
3.45 = x