^{3})+log (x

^{4}) / log(100x) = 5

log (x^{3})+log (x^{4}) / log(100x) = 5

x=

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Hi Dalia, so here it goes

Steps

log(x^{3}) + log(x^{4})/log(100x) = 5 Multiply log(100x) on both sides

log(x^{3}) + log(x^{4}) = 5log(100x) logM + logN = logMN, M = x^{3}, N = x^{4}

log[(x^{3})(x^{4}) = 5log(100x) Add exponents on left hand side

log(x^{3+4})=5log(100x) rlogM = logM^{r}

log(x^{7}) = log(100x)^{5} Bring 100x to the 5th power using (a^{s})^{t} = a^{s x t}

log(x^{7}) = log10,000x^{5} If logM = logN, then M=N, M=x^{7}, N=10,000x^{5}

x^{7} = (1x10^{10})x^{5} Set equation to 0 by subtracting (1x10^{10})x^{5} on both sides

x^{7} - (1x10^{10})x^{5} = 0 Take out Greatest Common Factor(GCF) of x^{5}

x^{5}(x^{2} - (1x10^{10})) = 0 Set each part equal to 0 and solve

x^{5} = 0 ; x^{2} - (1x10^{10}) = 0

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

x = 0 ; (x^{2})^{1/2}=(1x10^{10})^{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

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