Logarithm Worksheet - DOC by cwu19101

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```									                                Logarithm Worksheet

Definition
For all positive numbers a, where a ≠ 1

y = loga x is equivalent to ay = x.

A logarithm is an exponent, and loga x is the exponent to which a must be raised to
obtain x. The value of x will always be positive and the number a is called the base.

Common logarithms
If the base is not written, it is assumed to be base 10 and is called a common
logarithm. For example log 7 means log10 7.

Changing logarithmic equations into exponential equations (and back)
You must be able to convert an exponential equation into an equivalent logarithmic
equation and vise versa.

32 = 9                            log39 = 2

50 = 1                            ________________

___________                       log 1,000 = 3

( ¾ )-1 = 4/3                     ________________

___________                       log81 9 = ½

Natural logarithms
A number often used as a base for logarithms in many business and scientific
applications is the irrational number 2.71817… represented by the letter e. Instead
of writing logex, we abbreviate it as ln x. It is read as the natural log of x.

e3 = 20.0855…                     ln 20.0855… = 3

e7 = 1096.633…                    ________________

______________                    ln .1353… = -2

Your calculator can find the value for a logarithm if only if it is base 10 or base e
(common logs or natural logs). Use your calculator to find the value of each of the
following to 3 decimal places:
log 1000 = _________                  ln 16 = _________

log 16 = ___________                  ln e = __________

log 348 = __________                  ln .12 = _________

log .25 = __________                  ln e3 = __________

log 1 = ____________                  ln 1 = ___________

Properties of logarithms
From the definition of logarithms, you can find that

loga 1 = 0                     loga ak = k                  alogak = k , (k>0)

Rewrite the first 2 of the above equations as an exponential equation and the third
one as a logarithmic equation to confirm the property.

______________                 ______________               _______________

Since logarithms are exponents and we add exponents when we multiply terms,

loga xy = loga x + loga y

Likewise loga x/y = loga x - loga y            and loga xr = r loga x.

See problems on the next page.

Change of base
Since your calculator can only find values for base 10 and base e, you must convert
other bases into one of these.

loga x         =      logb x            =      log x           =     ln x
logb a                   log a                 ln x

For example log5 17 =          log 17          or      ln 17
log 5                   ln 5

Find the value of:

log5 17 =                                      log2.9 7.5 =

log32.5 =                                      log ½ 3 =

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