對數公式

若\;a>0,\;a≠1，\;c>0,\;c≠1\;
\;x>0,y>0，\;n \; 為任意實數

(3)\;log_a{xy}=log_ax+log_ay

(4)\;log_a{\frac{x}{y}}=log_ax-log_ay

(5)\;log_ax^n=nlog_ax

(1)\;log_aa=1，\;log_a1=0

(2)\;a^{log_ax}=x，\;log_aa^x=x

(6)\;log_ab=\frac{log_cb}{log_ca}

證明：

∵a^1=a

∴log_aa=1

∵a^0=1

∴log_a1=0

∵\fgcolor{255,153,204}{a}^\fgcolor{255,153,0}{x}=a^x

∵log\fgcolor{255,153,204}{_a}\fgcolor{255,153,0}{x}=log_ax

∴\fgcolor{255,153,204}{a}^{log_ax}=\fgcolor{255,153,0}{x}

∴log\fgcolor{255,153,204}{_a}a^x=\fgcolor{255,153,0}{x}

設\;log_ax=s,\;log_ay=t

則：\;a^s=x,\;a^t=y

log_a{xy}=\
\;\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=\
\;\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=log_ax+log_ay

log_a{(a^s·a^t)}

log_a(a^{s+t})

s+t

log_a{\frac{x}{y}}=\
\;\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=\
\;\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=\
\;\
\;\;\;\;\;\;\;\;\;\;\;\;=log_ax-log_ay

log_a\frac{a^s}{a^t}

log_aa^{s-t}

s-t

設\;log_ax=s

則\;a^s=x

⇒x^n=(a^s)^n=a^{sn}

⇒log_ax^n=log_aa^{sn}

=sn=nlog_ax

設\;log_ca=x,log_cb=y

⇒c^x=a,\;c^y=b

⇒c=a^{\frac{1}{x}}

⇒c^y=\left( a^{\frac{1}{x}}  \right) ^y=a^\frac{y}{x}=b

⇒log_aa^{\frac{x}{y}}=log_ab

⇒log_ab=\frac{x}{y}=\frac{log_cb}{log_ca}

Forward

s = 0 (slider from 0 to 40, step 1)
A = (-1.5507813477037053, 5.132754413501973)
text8: "" + (LaTeX[text2]) + ""
text14: "" + (LaTeX[text3]) + ""
text15: "" + (LaTeX[text9]) + ""
text20: "" + (LaTeX[text4]) + ""
B = (29.473720659229592, 1.0980139138133818)
C = (29.668644570919984, -3.0022671583915734)
D = (29.863568482610376, -7.4904126563456455)
text24: "" + (LaTeX[text5]) + ""
text25: "" + (LaTeX[text21]) + ""
text26: "" + (LaTeX[text22]) + ""
E = (28.9724877434543, 1.3750599322056085)
F = (29.000334016552927, -4.719952472423379)
G = (29.30664302063783, -9.5128485906089)
text28: "" + (LaTeX[text6]) + ""
text34: "" + (LaTeX[text7]) + ""