Question 378815
Hi Kenneth,


You might want to visit my website and take a look at my FREE VIDEOS of me teaching LOGARITHMS in my own class a few years ago.  It looks like you are SOLVING EXPONENTIAL EQUATIONS.  For a complete explanation of LOGARITHMS, please see my own website.  Click on my tutor name "Rapaljer" anywhere in algebra.com, and then click on my website that is given there.  Look for the link "Basic, Intermediate, and College Algebra: One Step at a Time" and select "College Algebra", "Chapter 4".  My explanations were written for students who have trouble understanding math, and according to my own students, much easier to understand than traditional textbooks.  


I also have TWO complete videos from my classes before I retired in which I taught the lessons myself.  The videos are FREE.  Look for "Rapalje Videos in Living Color" on my Homepage.


Now, let me solve the problems that I have time for:

{{{5^x=125^2^(x-1)}}}


Since both base numbers are powers of 5, there is an easy way to do this:
{{{5^x=5^3^2^(x-1)}}}


When you raise a power to a power, you must MULIPLY the exponents:
{{{5^x=5^(3*2*(x-1))}}} 
{{{5^x=5^(6x-6)}}}


Since the base numbers are equal, the exponents must be equal, so
{{{x=6x-6}}}

{{{-5x=-6}}}
{{{x=6/5}}}


In your second problem, in order to "undo" the e^, you should take the ln of each side of the equation:
{{{e^(4x)=9}}}
{{{ln e^(4x)= ln 9}}}

{{{4x=ln 9}}}
{{{x=(ln 9)/4}}}


Last problem, I'll help you set it up, but I don't have time to calculate it.


{{{3^(2x+1)=7^(x-1)}}}


Start by taking the ln of each side:
{{{ln(3^(2x+1))=ln(7^(x-1))}}}


By law of logarithms:
{{{(2x+1)*ln 3 = (x-1)*ln 7}}}


By distributive property:
{{{ 2x*ln3 + 1*ln3 = x*ln7-1*ln7}}}


Get all the x terms on one side by subtracting x* ln7 from each side.  And get all the NON-x terms to the right side by subtracting 1*ln3 from each side:
{{{2x*ln3 - x*ln7 = -1*ln7 - 1*ln3}}}

Next, factor out the x on the left side, and divide by the resulting factor on the left side:
{{{x= (-1*ln7 - 1*ln3)/(2*ln3-ln7)}}}


I think this is correct.  I don't have time to check it, but I have a TON of these on my website on the page called "Solving Exponential Equations."  Hang in there and good luck on your test!!


Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus