Question 563657
use natural logarithms to solve the equation 7 (3^z) = 19 (15^z) for z (three decimal places)
:
{{{7 (3^z) = 19 (15^z)}}}
divide both sides by 7
{{{3^z = (19/7) 15^z}}}
Divide both sides by 15^z
{{{3^z/15^z = 19/7}}}
Which is
{{{(3/15)^z = 19/7}}}
Reduce the fraction
{{{(1/5)^z = 19/7}}}
using nat logs
{{{ln((1/5)^z) = ln(19/7)}}}
log equiv of exponents
{{{z*ln(1/5) = ln(19/7)}}}
find the ln
-1.6094z = .9985
z = {{{.9985/(-1.6094)}}}
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