SOLUTION: Evaluate the following expressions: log3 81 log5 1/25 ln e^4 log.01

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Evaluate the following expressions: log3 81 log5 1/25 ln e^4 log.01      Log On


   

Question 172949: Evaluate the following expressions:
log3 81
log5 1/25
ln e^4
log.01
Found 2 solutions by solver91311, Earlsdon:
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
The rule you need to apply in each case is log%28b%2C%28y%29%29=x if and only if b%5Ex=y.

I'll show you the first one:

log%283%2C%2881%29%29=x if and only if 3%5Ex=81. So what does x have to be so that 3%5Ex=81? Since 3%2A3%2A3%2A3=3%5E4=81, x=4.

Work the rest of them the same way. Note that ln%28x%29=log%28e%2C%28x%29%29 (e is the base of the natural logarithms) and log%28x%29 (no base specified) is the same as log%2810%2C%28x%29%29

You might also want to make use of the rule log%28b%2C%28x%5En%29%29=n%2Alog%28b%2C%28x%29%29 and remember that x%5E1=x for all real x.

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Evaluate: Remember, "the logarithm of a number is the power to which the base must be raised to equal the number".
Log%5B3%5D%2881%29+=+x Rewrite in exponential form:
3%5Ex+=+81 Substitute:81+=+3%5E4
3%5Ex+=+3%5E4 The bases (3) are equal, so the exponents are equal.
highlight%28x+=+4%29
-------------------------
Log%5B5%5D%281%2F25%29+=+x Rewrite in exponential form:
5%5Ex+=+1%2F25 Substitute:1%2F25+=+1%2F5%5E2=5%5E%28-2%29
5%5Ex+=+5%5E%28-2%29 The bases are equal, so the exponents are equal.
highlight%28x+=+-2%29
-------------------------
ln%28e%5E4%29+=+x Apply the "power rule" for logarithms:
4%2Aln%28e%29+=+x Substitute ln%28e%29+=+1
4+=+x or highlight%28x+=+4%29
-------------------------
Log%280.01%29+=+x Rewrite in exponential form:
10%5Ex+=+0.01 Substitute 0.01+=+1%2F100=1%2F10%5E2+=+10%5E%28-2%29
10%5Ex+=+10%5E%28-2%29 The bases are equal, so the exponents are equal.
highlight%28x+=+-2%29