SOLUTION: use natural logarithms to solve the equation 7 (3^z) = 19 (15^z) for z (three decimal places)

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Question 563657: use natural logarithms to solve the equation 7 (3^z) = 19 (15^z) for z (three decimal places)
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
use natural logarithms to solve the equation 7 (3^z) = 19 (15^z) for z (three decimal places)
:
7+%283%5Ez%29+=+19+%2815%5Ez%29
divide both sides by 7
3%5Ez+=+%2819%2F7%29+15%5Ez
Divide both sides by 15^z
3%5Ez%2F15%5Ez+=+19%2F7
Which is
%283%2F15%29%5Ez+=+19%2F7
Reduce the fraction
%281%2F5%29%5Ez+=+19%2F7
using nat logs
ln%28%281%2F5%29%5Ez%29+=+ln%2819%2F7%29
log equiv of exponents
z%2Aln%281%2F5%29+=+ln%2819%2F7%29
find the ln
-1.6094z = .9985
z = .9985%2F%28-1.6094%29
z = -.6206
:
:
Check solution on a calc:
enter 7*3^-.62) = 3.54
enter 19*15^-.62 = 3.54; confirms our solution of z = -.6206