SOLUTION: Use the change of base formula to evaluate {{{log(2,17)}}}

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Question 311598: Use the change of base formula to evaluate log%282%2C17%29
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Use the change of base formula to evaluate log%282%2C17%29

Change of base formula:

log%28OLD_BASE%2C%28X%29%29%22=%22log%28NEW_BASE%2C%28X%29%29%2Flog%28NEW_BASE%2C%28OLD_BASE%29%29

The OLD_BASE is 2 and the NEW_BASE is either 10 or e, take your pick:

If we pick 10 for the NEW_BASE, then

log%28OLD_BASE%2C%28X%29%29%22=%22log%28NEW_BASE%2C%28X%29%29%2Flog%28NEW_BASE%2C%28OLD_BASE%29%29
becomes
log%282%2C%2817%29%29%22=%22log%2810%2C%2817%29%29%2Flog%2810%2C%282%29%29

When the base is 10 we don't write the 10 but leave it understood:
log%282%2C%2817%29%29%22=%22log%28%2817%29%29%2Flog%28%282%29%29

Then we cet a calculator and find those logarithms with the LOG key:
log%282%2C%2817%29%29%22=%221.230448921%2F0.3010299957=4.08746284
--------------------------------------------
If we pick e for the NEW_BASE, then

log%28OLD_BASE%2C%28X%29%29%22=%22log%28NEW_BASE%2C%28X%29%29%2Flog%28NEW_BASE%2C%28OLD_BASE%29%29
becomes
log%282%2C%2817%29%29%22=%22log%28e%2C%2817%29%29%2Flog%28e%2C%282%29%29

When the base is e we don't write log%28e%2C%22%22%29 but write ln%28%22%22%29 instead:
log%282%2C%2817%29%29%22=%22ln%2817%29%2Fln%282%29

Then we cet a calculator and find those natural logarithms with the LN key:
log%282%2C%2817%29%29%22=%222.833213344%2F0.6931471806=4.087462841

Notice that regardless of whether we picked 10 or e for the NEW_BASE,
we ended up with the same answer.

Edwin