Question 41576
<pre><font size = 5><b>Solve the following equations:

a) 5^(2x+1) = 48     b) log6(x+41)-log4(x+1)=2


   5<sup>2x+1</sup> = 48

Take logs of both sides

  log(5<sup>2x+1</sup>) = log(48)

Use the rule:   log(A<sup>B</sup>) = B·log(A) on the left side

 (2x+1)log(5) = log(48)

Divide both sides by log(5)

 (2x+1)log(5)    log(48)
 ———————————— =  —————————
    log(5)        log(5)

         1 
 (2x+1)<s>log(5)</s>    log(48)
 ———————————— = —————————
    <s>log(5)</s>        log(5)
      1

                 log(48)
       2x + 1 = —————————
                  log(5) 

                 log(48)
           2x = ————————— - 1
                  log(5)


                 log(48)     log(5)
           2x = ————————— - —————————
                  log(5)     log(5)

                 log(48) - log(5)
           2x = ——————————————————
                      log(5) 


Divide both sides by 2:

                 log(48) - log(5)
            x = ——————————————————
                    2·log(5) 

                 1.681241237 - 0.6989700043
            x = ————————————————————————————
                    2·(0.6989700043)

                 0.982271233
            x = —————————————
                 1.397940009

            x = .7026562134

===============================================

 b) 

       log6(x+41) - log4(x+1) = 2
     
Use rule:   

log(A) - log(B) = log(A/B) on left side

                    <font face = "symbol">æ</font>6(x+41)<font face = "symbol">ö</font>	
                log <font face = "symbol">ç</font>———————<font face = "symbol">÷</font> = 2
                    <font face = "symbol">è</font> 4(x+1)<font face = "symbol">ø</font> 
                       
Use the rule: log(A) = B is equivalent to
the equation A = 10<sup>B</sup>  

                      6(x+41)
                     ———————— = 10<sup>2</sup>
                       4(x+1)  

                      3
                      <s>6</s>(x+41)
                     ———————— = 10<sup>2</sup>
                       <s>4</s>(x+1)   
                       2
             
                      3(x+41)
                     ———————— = 100
                       2(x+1)

Multiply both sides by 2(x+1)

                      3(x+41) = 100·2(x+1)

                      3(x+41) = 200(x+1)

                     3x + 123 = 200x + 200            

                        -197x = 77

                            x = -77/197

Edwin McCravy
AnlytcPhil@aol.com</pre>