Question 4694
1.  ln7 + ln(x-3) = ln(x + 15)

First get the variables on one side (the right side) by subtracting ln(x-3) from each side.
ln 7 = ln(x+15) - ln(x-3)


Use the property of logs to express the right side as a log of a single quantity: {{{ ln M - ln N = ln (M/N) }}}
{{{ln 7 = ln ((x+15)/(x-3)) }}}


Because ln A = ln B, you can say that A=B.  In the same way,
{{{7 = (x+15)/(x-3) }}}


Multiply both sides of the equation by (x-3):
{{{7(x-3) = (x+ 15) }}}
{{{7x - 21 = x + 15 }}}
{{{6x = 36}}}
{{{x=6}}}


Check to make sure x= 6 doesn't give you the log of a negative.  It does not.



2.  {{{28 = 19 + 12^(x-7)}}}

Subtract 19 from each side of the equation:
{{{9 = 12^(x-7)}}}


Take the ln of each side:
{{{ln (9) = ln (12^(x-7)) }}}


By law of logarithms:
{{{ln (9) = (x-7) * ln (12)}}}


By distributive property:
{{{ln (9) = x ln(12) - 7 ln(12) }}}


Add 7 ln (12) to each side:
{{{ln (9) + 7 ln (12) = x ln (12) }}}


Divide both sides by ln (12)

{{{ (ln (9) + 7 ln (12))/ (ln (12 )) = x }}}


I'm very sorry, that's all I have time for.  Maybe someone else will do the others, or perhaps you can repost them.


R^2 at SCC