[tex]\frac{log_{2}(24)}{log_{96}(2)} - \frac{log_{2}(192)}{log_{12}(2)}[/tex]
log₂(24): [tex]2^{x} = 24[/tex]
[tex]ln(2^{x}) = ln(24)[/tex]
[tex]xln(2) = ln(24)[/tex]
[tex]x = \frac{ln(24)}{ln(2)}[/tex]
[tex]x = \frac{ln(8 * 3)}{ln(2)}[/tex]
[tex]x = \frac{ln(8) + ln(3)}{ln(2)}[/tex]
[tex]x = \frac{3ln(2) + ln(3)}{ln(2)}[/tex]
[tex]x = \frac{3ln(2)}{ln(2)} + \frac{ln(3)}{ln(2)}[/tex]
[tex]x = 3 + 1.1[/tex]
[tex]x = 4.1[/tex]
log₉₆(2): [tex]96^{x} = 2[/tex]
[tex]ln(96^{x}) = ln(2)[/tex]
[tex]xln(96) = ln(2)[/tex]
[tex]xln(32 * 3) = ln(2)[/tex]
[tex]x(ln(32) + ln(3)) = ln(2)[/tex]
[tex]\frac{x(ln(32) + ln(3))}{ln(32) + ln(3)} = \frac{ln(2)}{ln(32) + ln(3)}[/tex]
[tex]x = \frac{ln(2)}{ln((2)^{5}) + ln(3)}[/tex]
[tex]x = \frac{ln(2)}{5ln(2) + ln(3)}[/tex]
[tex]x = 0.7[/tex]
[tex]\frac{log_{2}(24)}{log_{96}(2)} = \frac{3 + \frac{ln(3)}{ln(2)}}{\frac{ln(2)}{5ln(2) + ln(3)}} = \frac{4.1}{0.7} = 5\frac{6}{7}[/tex]
log₂(192): [tex]2^{x} = 192[/tex]
[tex]ln(2^{x}) = ln(192)[/tex]
[tex]xln(2) = ln(64 * 3)[/tex]
[tex]xln(2) = ln(64) + ln(3)[/tex]
[tex]xln(2) = ln((2)^{6}) + ln(3)[/tex]
[tex]xln(2) = 6ln(2) + ln(3)[/tex]
[tex]\frac{xln(2)}{ln(2)} = \frac{6ln(2) + ln(3)}{ln(2)}[/tex]
[tex]x = \frac{6ln(2)}{ln(2)} + \frac{ln(3)}{ln(2)}[/tex]
[tex]x = 6 + 1.1[/tex]
[tex]x = 7.1[/tex]
log₁₂(2): [tex]12^{x} = 2[/tex]
[tex]ln(12^{x}) = ln(2)[/tex]
[tex]xln(12) = ln(2)[/tex]
[tex]xln(4 * 3) = ln(2)[/tex]
[tex]x(ln(4) + ln(3)) = ln(2)[/tex]
[tex]\frac{x(ln(4) + ln(3))}{ln(4) + ln(3)} = \frac{ln(2)}{ln(4) + ln(3)}[/tex]
[tex]x = \frac{ln(2)}{ln((2)^{2}) + ln(3)}[/tex]
[tex]x = \frac{ln(2)}{2ln(2) + ln(3)}[/tex]
[tex]x = 0.3[/tex]
[tex]\frac{log_{2}(192)}{log_{12}(2)} = \frac{log_{2}(192)}{log_{12}(2)} = \frac{7.1}{0.3} = 23\frac{2}{3}[/tex]
[tex]5\frac{6}{7} - 23\frac{2}{3}[/tex]
[tex]-17\frac{17}{21}[/tex]
