Question

Lösen der Logarithmusgleichung: log_{5}(x) + ln(x^{3}) = lg(4x)

Answer 1

LOG5(x) + LN(x^3) = LOG10(4·x)

LN(x)/LN(5) + 3·LN(x) = (LN(4) + LN(x))/LN(10)

LN(x)/LN(5) + 3·LN(x) = LN(4)/LN(10) + LN(x)/LN(10)

LN(x)/LN(5) + 3·LN(x) - LN(x)/LN(10) = LN(4)/LN(10)

LN(x)*(1/LN(5) + 3 - 1/LN(10)) = LN(4)/LN(10)

LN(x) = LN(4)/LN(10) / (1/LN(5) + 3 - 1/LN(10)) = 0.1889088012

x = e^0.1889088012 = 1.207930785

Answer 2

lg(x) / lg(5)  +  lg(x^3) / lg(e)   =   lg(4x)  Danach wird es falsch, du hast lg(x/y)   mit lg(x) / lg(y) verwechselt !

lg(x) / lg(5)  + 3* lg(x) / lg(e)   =   lg(4) + lg(x)   |   - lg(x)

lg(x) / lg(5)  + 3* lg(x) / lg(e)   - lg( x)     =   lg(4)    Dann lg(x) ausklammern

lg(x)   *   (1 / lg(5)  + 3 / lg(e)   - 1 )     =   lg(4)       |  : Klammer

lg(x)   =      lg(4)    /   (1 / lg(5)  + 3 / lg(e)   - 1 )  

also  x =  10 hoch ( lg(4)    /   (1 / lg(5)  + 3 / lg(e)   - 1 )  )