Download Notes: Solving Logarithmic Equations

Notes: Solving Logarithmic Equations
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Date _________
Common Logarithm
1) __A common logarithm is a logarithm that uses base 10____________
ex: log10 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 a b 
log 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
__e0 = 1____________
ex 5: ln 4 = 1.386
__e1.386 = 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.
1
ex 6: log 4
=x
ex 7: ln 20 = x
2
log (1/2) = - 2
log (4)
x = 2.996
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
35 = 343
3 1/3
[x ]
x3 = 64
= (64) 1/3
x=4
ex 11: log 2 (x + 1) = 1
21 = x + 1
2=x+1
1=x
ex 13: log (3x + 1) = 5
105 = 3x + 1
100000 = 3x + 1
99999 = 3x
33333 = x
ex 15: log 5x + 3 = 9
log 5x + 3 = 9
log 5x = 6
106 = 5x
1,000,000 = 5x
200,000 = x
ex 12: log 2 x + 1 = 1
log 2 x + 1 = 1
log 2 x = 0
20 = x
1=x
ex 14: 3 log (2x – 2) = 12
log (2x – 2) = 4
104 = 2x + 2
10000 = 2x+ 2
10002 = 2x
5001 = x
ex 16: log (7 – 2x) = -1
log (7 – 2x) = -1
10-1 = 7 - 2x
1/10 = 7 – 2x
1/10 – 7 = - 2x
- 6.9 = - 2x
3.45 = x