(log_{10}(80))/(log_{10)(2x)}=(log_{10}(500))/(log_{10)(5x)}

Lösung

\frac{lo g _{10} ( 80 )}{lo g _{10} ( 2 x )} = \frac{lo g _{10} ( 500 )}{lo g _{10} ( 5 x )}

x = 1 0^{l o g_{\frac{25}{4}} (\frac{2 ^{4 l o g_{10} (5) - 2 l o g_{10} (2)}}{5 ^{- l o g_{10} (5) + 3 l o g_{10} (2)}})}

Dezimale

x = 4.47213 \dots

Schritte zur Lösung

\frac{lo g _{10} ( 80 )}{lo g _{10} ( 2 x )} = \frac{lo g _{10} ( 500 )}{lo g _{10} ( 5 x )}

Wende die Regeln für Multipikation bei Brüchen an: Wenn

\frac{a}{b} = \frac{c}{d}

dann

a \cdot d = b \cdot c

lo g_{10} (80) lo g_{10} (5 x) = lo g_{10} (2 x) lo g_{10} (500)

Wende die log Regel an

x = 1 0^{l o g_{\frac{25}{4}} (\frac{8 0 ^{l o g_{10} (5)}}{50 0 ^{l o g_{10} (2)}})}

Löse

x = 1 0^{l o g_{\frac{25}{4}} (\frac{8 0 ^{l o g_{10} (5)}}{50 0 ^{l o g_{10} (2)}})} : x = 1 0^{l o g_{\frac{25}{4}} (\frac{2 ^{4 l o g_{10} (5) - 2 l o g_{10} (2)}}{5 ^{- l o g_{10} (5) + 3 l o g_{10} (2)}})}

x = 1 0^{l o g_{\frac{25}{4}} (\frac{2 ^{4 l o g_{10} (5) - 2 l o g_{10} (2)}}{5 ^{- l o g_{10} (5) + 3 l o g_{10} (2)}})}

Überprüfe die Lösungen:

x = 1 0^{l o g_{\frac{25}{4}} (\frac{2 ^{4 l o g_{10} (5) - 2 l o g_{10} (2)}}{5 ^{- l o g_{10} (5) + 3 l o g_{10} (2)}})}

Wahr

Deshalb ist die Lösung

x = 1 0^{l o g_{\frac{25}{4}} (\frac{2 ^{4 l o g_{10} (5) - 2 l o g_{10} (2)}}{5 ^{- l o g_{10} (5) + 3 l o g_{10} (2)}})}

lo g_{5} ((x - 2)^{2}) \cdot lo g_{2} (5) = 2

20 lo g_{10} (x) = 28

ln^{2} (x) = ln (e)

2 lo g_{10} (x - 2) = 2 x - 7

lo g_{x} (64) + 3 \cdot lo g_{4} (x) - lo g_{x} (4) = 5