Question 30861
Melissa the first one is solved using natural log

Using your equation Cn=500(0.82)^n-1

{{{Cn < 150}}}
{{{150<500(0.82)^(n-1)}}}
{{{ln(150)<ln(500(0.82)^(n-1))}}} use the natural log function
{{{ln(150)<ln(500) + ln(0.82)^(n-1)}}} Product rule of logs
{{{ln(150)<ln(500) + n-1*ln(0.82)}}} Power rule of logs
{{{5.01<6.21-.20*(n-1)}}} Solve for Ln
{{{5.01<6.21-.20n+.20}}} distribute
{{{5.01<6.41-.20n}}} combine like terms
{{{-1.4<-.20n}}} subtract 6.41 from both sides
{{{7>n}}} divide by -.20 and flip the inequality


I will look at the second question and see what I can do now.


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VIA YOUR COMMENTS I AM REWRITING THIS SOLUTION:
The second question equations are
{{{A=3000(1+0.054/12)^(12*t)}}}
{{{A=2700(1+0.068/4)^(4*t)}}}

First set the two equations equal to each other:

{{{2700(1+0.068/4)^(4t)=3000(1+0.054/12)^(12t)}}}  Solve what you can by combining like terms
{{{2700(1.017)^(4t)=3000(1.0045)^(12t)}}} Now introduce the natural log
{{{ln(2700(1.017)^(4t))=ln(3000(1.0045)^(12t))}}} Use Product Rule of logs
{{{ln(2700)+ln(1.017^(4t))=ln(3000)+ln(1.0045^(12t))}}} Use Power Rule of Logs
{{{ln(2700)+(4t)*ln(1.017)=ln(3000)+(12t)*ln(1.0045)}}} Solve the ln portions
{{{7.90+(4t)*0.0169=8.01+(12t)*0.0045}}}  Distributive Property
{{{7.90+0.0676t=8.01+0.054t}}} Combine like terms
{{{0.0136t=0.11}}} Solve for t
{{{t=8.09years}}}

Looks like they aren't going to have the same amount of money for their trip!!