- \(\log _{a} 1=0\)
- \(\log _{a} a=1\)
- \(\log _{a}(b \cdot c)=\log _{a} b+\log _{a} c\)
- \(\log _{a}\left(\frac{b}{c}\right)=\log _{a} b-\log _{a} c\)
- \(\log _{a} b^{n}=n \cdot \log b\)
- \(\log _{a} b=\frac{\log _{c} b}{\log _{c} a}\)
EXEMPLE : Sabent que \(\log _{2} a=1,3\) i que \(\log _{2} b=0,4\), calcula el valor del següent logaritme, sense utilitzar la calculadora:
\( \log _{2} \frac{\sqrt{a^{3}} \cdot\left(a^{2} \cdot b\right)^{2}}{\sqrt[3]{a \cdot b}} \\=\)
\(= \log _{2} \frac{\sqrt{a^{3}} \cdot\left(a^{2} \cdot b\right)^{2}}{\sqrt[3]{a \cdot b}}=\log _{2}\left(\sqrt{a^{3}} \cdot\left(a^{2} \cdot b\right)^{2}\right)-\log _{2}(\sqrt[3]{a \cdot b}) \\=\)
\( =\left(\log _{2} \sqrt{a^{3}}\right)+\left(\log _{2}\left(a^{2} \cdot b\right)^{2}\right)-\log _{2}(\sqrt[3]{a \cdot b}) \\=\)
\(=\log _{2} a^{\frac{3}{2}}+2 \cdot \log _{2}\left(a^{2} \cdot b\right)-\log _{2}(a \cdot b)^{\frac{1}{3}} \\=\)
\( =\frac{3}{2} \cdot \log _{2} a+2 \cdot\left(\log _{2} a^{2}+\log _{2} b\right)-\frac{1}{3} \cdot \log _{2}(a \cdot b) \\=\)
\( =\frac{3}{2} \cdot \log _{2} a+2 \cdot\left(2 \cdot \log _{2} a+\log _{2} b\right)-\frac{1}{3} \cdot\left(\log _{2} a+\log _{2} b\right) \\=\)
\(=\left(\frac{3}{2}+4-\frac{1}{3}\right) \cdot \log _{2} a+\left(2-\frac{1}{3}\right) \cdot \log _{2} b=\frac{31}{6} \cdot \log _{2} a+\frac{5}{3} \cdot \log _{2} b \\=\frac{31}{6} \cdot 1,3+\frac{5}{3} \cdot 0,4=7,38\)