PROPERTIES OF LOGARITHMIC FUNCTIONS
EXPONENTIAL FUNCTIONS
An exponential function is a function of the form f ( x ) = b x , where b > 0 and x is any real
number. (Note that f ( x ) = x 2 is NOT an exponential function.)
LOGARITHMIC FUNCTIONS
log b x = y means that x = b y where x > 0, b > 0, b ≠ 1
Think: Raise b to the power of y to obtain x. y is the exponent.
The key thing to remember about logarithms is that the logarithm is an exponent!
The rules of exponents apply to these and make simplifying logarithms easier.
Example: log 10 100 = 2 , since 100 = 10 2 .
log 10 x is often written as just log x , and is called the COMMON logarithm.
log e x is often written as ln x , and is called the NATURAL logarithm (note: e ≈ 2.718281828459... ).
PROPERTIES OF LOGARITHMS
EXAMPLES
1. log b MN = log b M + log b N
log 50 + log 2 = log 100 = 2
Think: Multiply two numbers with the same base, add the exponents.
M
 56 
= log b M − log b N
log 8 56 − log 8 7 = log 8   = log 8 8 = 1
N
 7
Think: Divide two numbers with the same base, subtract the exponents.
2. log b
3. log b M P = P log b M
log 100 3 = 3 ⋅ log 100 = 3 ⋅ 2 = 6
Think: Raise an exponential expression to a power and multiply the exponents together.
log b b x = x
log b 1 = 0 (in exponential form, b 0 = 1 )
log b b = 1
log 10 10 = 1
ln e = 1
log b b = x
log 10 10 = x
ln e x = x
b logb x = x
Notice that we could substitute y = log b x into the expression on the left
x
x
ln 1 = 0
to form b y . Simply re-write the equation y = log b x in exponential form
as x = b y . Therefore, b logb x = b y = x .
Ex: e ln 26 = 26
CHANGE OF BASE FORMULA
log b N =
log a N
, for any positive base a.
log a b
log 12 5 =
log 5 0.698970
≈
≈ 0.6476854
log 12 1.079181
This means you can use a regular scientific calculator to evaluate logs for any base.
1.
2.
3.
4.
Practice Problems contributed by Sarah Leyden, typed solutions by Scott Fallstrom
Solve for x (do not use a calculator).
2
log 9 (x − 10 ) = 1
6. log 3 27 x = 4.5
10. log 2 x 2 − log 2 (3x + 8) = 1
3
log 3 3 2 x +1 = 15
7. log x 8 = −
11. ( 1 2 ) log 3 x − ( 13 ) log 3 x 2 = 1
2
log x 8 = 3
8. log 6 x + log 6 ( x − 1) = 1
log 5 x = 2
1
9. log 2 x 2 + log 2 ( 1x ) = 3
log (x 2 − 7 x + 7 ) = 0
5.
5
Solve for x, use your calculator (if needed) for an approximation of x in decimal form.
12. 7 x = 54
15. 10 x = e
18. 8 x = 9 x
13. log 10 x = 17
16. e − x = 1.7
19. 10 x +1 = e 4
17. ln (ln x ) = 1.013
20. log x 10 = −1.54
14. 5 x = 9 ⋅ 4 x
Solutions to the Practice Problems on Logarithms:
1. log 9 (x 2 − 10) = 1 ⇒ 91 = x 2 − 10 ⇒ x 2 = 19 ⇒ x = ± 19
2. log 3 3 2 x +1 = 15 ⇒ 315 = 3 2 x +1 ⇒ 2 x + 1 = 15 ⇒ 2 x = 14 ⇒ x = 7
3. log x 8 = 3 ⇒ x 3 = 8 ⇒ x = 2
4. log 5 x = 2 ⇒ 5 2 = x ⇒ x = 25
5. log 5 (x 2 − 7 x + 7 ) = 0 ⇒ 5 0 = x 2 − 7 x + 7 ⇒ 0 = x 2 − 7 x + 6 ⇒ 0 = ( x − 6)( x − 1) ⇒ x = 6 or x = 1
( )
6. log 3 27 x = 4.5 ⇒ log 3 33
x
= 4.5 ⇒ log 3 33 x = 4.5 ⇒ 3 x = 4.5 ⇒ x = 1.5
7. log x 8 = − 32 ⇒ x − 2 = 8 ⇒ x = 8 − 3 ⇒ x =
3
2
1
4
log 6 x + log 6 ( x − 1) = 1 ⇒ log 6 (x 2 − x ) = 1 ⇒ x 2 − x = 6 ⇒ x 2 − x − 6 = 0 ⇒
8.
(x − 3)(x + 2) = 0 ⇒ x = 3
or x = −2. Note : x = −2 is an extraneous solution, which solves only
the new equation. x = 3 is the only solution t o the original equation.
x 2
1
9. log 2 x + log 2   = 3 ⇒ log 2 
 x
 x

1
1
 = 3 ⇒ log 2 x − 2 = 3 ⇒ 2 3 = x − 2 ⇒ x = 2 3


2
2
x2
log 2 x − log 2 (3 x + 8) = 1 ⇒ log 2 3 x +8 = 1 ⇒ 3 xx+8 = 2 ⇒ x 2 = 6 x + 16 ⇒
( )
1
1
10.
11.
2
−2
( )
x 2 − 6 x − 16 = 0 ⇒ ( x − 8)( x + 2 ) = 0 ⇒ x = 8 or x = −2
(
x 2
1
2
= 1 ⇒ log 3  2 3  = 1 ⇒ x 2 − 3 = 3 ⇒
2 ) log 3 x − ( 3 ) log 3 x = 1 ⇒ log 3 x − log 3 x
x 
1
1
1
2
1
x − 6 = 3 ⇒ x = 3−6 =
1
5x
4x
3
1
729
log 54
≈ 2.0499
13. log 10 x = 17 ⇒ x = 1017
log 7
12. 7 x = 54 ⇒ x = log 7 54 ⇒ x =
14. 5 x = 9 ⋅ 4 x ⇒
2
2
= 9 ⇒ ( 54 ) = 9 ⇒ x = log 5 9 ⇒ x ≈ 9.8467
x
4
15. 10 = e ⇒ x = log 10 e ⇒ x = log e ≈ 0.4343
x
16. e − x = 1.7 ⇒ − x = ln 1.7 ⇒ x = − ln 1.7 ≈ −0.5306
1.013
17. ln (ln x ) = 1.013 ⇒ ln x = e1.013 ⇒ x = e e ≈ 15.7030
18. 8 x = 9 x ⇒ 1 = ( 98 ) ⇒ x = log 9 1 ⇒ x = 0
x
8
19. 10
x +1
= e ⇒ x + 1 = log e ⇒ x = log e 4 − 1 = log e 4 − log 10 ⇒ x = log
4
4
20. log x 10 = −1.54 ⇒ x −1.54 = 10 ⇒ x = 10 − 1.54 ≈ 0.2242
1
( ) ≈ 0.7372
e4
10
=
1
64