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Logarithms

We write the logarithm with base 10 as log(x) and that with base e as ln(x). These are just the inverse functions for the corresponding exponentials, so that

if y = log(x) then 10^y = x
if y = ln(x) then e^y = x

In words, log(x) is the exponent of 10 that gives x, and ln(x) is the exponent of e that gives x, that is,

10^(log(x)) = x and e^(ln(x)) = x

All logarithm functions have some useful properties:

Properties of logarithms
(M and N are positive.)
log(MN) = log(M) + log(N)   ln(MN) = ln(M) + ln(N) logarithm of a product
log(M/N) = log(M) - log(N)   ln(M/N) = ln(M) - ln(N) logarithm of a quotient
log(M^p) = p log(M)   ln(M^p) = p ln(M) logarithm of a power
log(1) = 0   ln(1) = 0 logarithm of +1
log(10) = 1   ln(e) = 1 logarithm of the base
Be careful not to make the mistake of trying to use these properties when they do not apply. None of the following forms can be rewritten using the properties above.
log(M+N) ln(M+N) no rules for the logarithm of a sum
log(M)/log(N) ln(M)/ln(N) no rules for the quotient of logarithms

Relationship between logarithms and exponentials

These follow from the definition of the logarithms as the inverse functions for the corresponding exponential functions.

Logarithms and exponentials
log(N) = x <=> 10^x = N   ln(N) = x <=> e^x = N
log(10^x) = x   ln(e^x) = x
10^{log(x)} = x   e^{ln(x)} = x
Examples
For each of these, note what properties of logarithms are being used.
  1. log(10,000) = log(10^4) = 4 (the power of 10 needed to get 10,000 is 4).
  2. ln(1)=0 (the power of e needed to get 1 is 0).
  3. log(x) = -3 means 10^(-3) = x
  4. ln(x) = sqrt(2) means e^(sqrt(2)) = x
  5. 10^3 = 1000 means log(1000) = 3
  6. 10^(-2) = 0.01 means log(0.01) = -2
  7. log(10x) = log(10) + log(x) = 1 + log(x)
  8. ln(e^2/sqrt(x)) = ln(e^2) - ln(sqrt(x)) = 2ln(e) - ln(x^(1/2)) = 2*1 - (1/2) ln(x) = 2 - (1/2) ln(x)
  9. 3( 2log(M) + (4/3)log(N) ) = 6 log(M) + 4 log(N) = log(M^6) + log(N^4) = log(M^6 N^4)
  10. e^(3ln(x)) + 2ln(e^(-x)) = e^(ln(x^3)) + 2(-x) = x^3 - 2x
  11. log(10^(2x)) + log(sqrt(10)) = 2x log(10) + log(10^(1/2)) = 2x + (1/2)log(10) = 2x + 1/2
Practice

Be sure you've gone through each step in the examples above before doing these. Once you've worked them until you're sure that you understand them, go on to the next section. There are more problems of this type in the section test at the end of the section.

Note that you can get new practice problems by clicking the "Refresh" button at the bottom of the practice set.

  1. Rewrite the following expression in the form
    a + b ln(x)
    ln (  
    e ^a x
    		  )   =     +   ln(x)
    .
    ( e x ) ^b
    (Enter answers, then click: . Answer message: )
  2. Rewrite the following expression in the form
    ln(c)
    ln (   c   )   +   ln (  
    d
    .
    2
    		  )   -   ln (   g ^h   )   =   ln ( )
    (Enter answer, then click: . Answer message: )
  3. Rewrite the following expression in the form
    a + b log(x)
    log (   10 ^j x ^k   )   -   log (   x ^m   )   =     +   log(x)
    (Enter answers, then click: . Answer message: )
For more practice, click: .
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precal: 6.1 - logarithms
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