find the exact solution to the equation below

Question

log (x³)+log (x⁴) / log(100x) = 5

x=

Answer 1

Hi Dalia, so here it goes

                                                                            Steps

log(x³) + log(x⁴)/log(100x) = 5                      Multiply log(100x) on both sides                          

log(x³) + log(x⁴) = 5log(100x)                       logM + logN = logMN, M = x³, N = x⁴

log[(x³)(x⁴) = 5log(100x)                              Add exponents on left hand side

log(x³⁺⁴)=5log(100x)                                    rlogM = logM^r

log(x⁷) = log(100x)⁵                                     Bring 100x to the 5th power using (a^s)^t = a^{s x t}

log(x⁷) = log10,000x⁵                                   If logM = logN, then M=N, M=x⁷, N=10,000x⁵

x⁷ = (1x10¹⁰)x⁵                                           Set equation to 0 by subtracting (1x10¹⁰)x⁵ on both sides

x⁷ - (1x10¹⁰)x⁵ = 0                                           Take out Greatest Common Factor(GCF) of x⁵

x⁵(x² - (1x10¹⁰)) = 0                                         Set each part equal to 0 and solve

x⁵ = 0                      ;      x² - (1x10¹⁰) = 0

(x⁵)^1/5 = (0)^1/5        ;            ^{+1x10^10      1x10^10}  

x = 0                       ;       (x²)^1/2=(1x10¹⁰)^1/2

                                        x = ±100,000

So, x =0,100

Now Check your solution

x = 0 => log(0)³ + log(0)⁴/log(100(0) = 5              

              You cannot do the log0, because the there is a logarthms are undefined at x = 0 where there is  .             a vertical asymptote at y=0   

x = -100,000, logarthms cannot be negative

x=100,00 => log(100,000)³ + log(100,000)⁴/log(100)(100,000) = 5

                     => 15+20/7 = 5

                     =>  35/7 = 5

                     =>       5 = 5

The exact solution is x = 100,000

 

Happy to help, feel free to rate me

Shannon