Lanjutan Materi Fungsi Logaritma (Kelas X Matematika Peminatan)

 MENGINGAT KEMBALI

$\text{E. Sifat-Sifat Eksponen dan Logaritma}$

$\text{E.1 Persamaan Eksponen dan Logaritma}$

$\begin{array}{|ll|}\hline \text{Eksponens}& \text{Logaritma}\\ {\displaystyle a^n=\underset{nfaktor}{\underbrace{a\times a\times a\times \cdots \times a}}}& {}^a\log b=c\Rightarrow a^c=b\\ \bullet a^p\times a^q=a^{p+q}& \bullet {}^a\log x+{}^a\log y={}^a\log xy\\ \bullet a^p:a^q=a^{p-q}& \bullet {}^a\log x-{}^a\log y={}^a\log {\displaystyle \frac{x}{y}}\\ \bullet {\left(a^p\right)}^q=a^{p.q}& \bullet {}^a\log x={\displaystyle \frac{{}^m\log x}{{}^m\log a}}\\ \bullet {\displaystyle \sqrt[q]{a^p}={\displaystyle a^{\left(\frac{p}{q}\right)}}}& \bullet {}^a\log b\times {}^b\log c={}^a\log c\\ \bullet {(a\times b)}^p=a^p\times b^p& \bullet {}^{a^m}\log b^n={\displaystyle \frac{n}{m}\times {}^a\log b}\\ \bullet {\left({\displaystyle \frac{a}{b}}\right)}^p={\displaystyle \frac{{\displaystyle a^p}}{{\displaystyle b^p}}}& \bullet {\displaystyle a^{{}^a\log b}=b}\\ \bullet a^{-p}={\displaystyle \frac{1}{{\displaystyle a^p}}}& \bullet {}^a\log b={\displaystyle \frac{1}{{}^b\log a}}\\ \bullet a^0=1,a\ne 0& \bullet {}^a\log 1=0\\ \bullet a^1=1& \bullet {}^a\log a=1\\ \{\begin{array}{l}a,b\in \mathbb{R}\\ p,q\in \mathbb{Q}\end{array}& \{\begin{array}{ll}a\ne 0& \\ a>0& (\text{bilangan pokok})\\ x,y>0& (\text{numerus})\end{array}\\ \hline\end{array}$

Selanjutnya

$\begin{array}{|lll|}\hline \text{No}& \text{Bentuk}& \text{Syarat}\\ 1.& a^{f(x)}=1& a\ne 0,\text{maka}f(x)=0\\ 2.& a^{f(x)}=a^p& a>0,a\ne 1,\text{maka}f(x)=p\\ 3.& a^{f(x)}=a^{g(x)}& a>0,a\ne 1,\text{maka}f(x)=g(x)\\ 4.& a^{f(x)}=b^{f(x)}& a\ne 0,b\ne 0,\text{maka}f(x)=0\\ 5.& f(x{)}^{g(x)}=1& \{\begin{array}{ll}f(x)=1& \\ g(x)=0,& \text{jika}f(x)\ne 0\\ f(x)=-1,& \text{jika}g(x)=\text{genap}\end{array}\\ 6.& f(x{)}^{g(x)}=f(x{)}^{h(x)}& \{\begin{array}{ll}(i).g(x)=h(x)& \\ (ii).f(x)=1& \\ (iii).f(x)=0,& g(x)>0,h(x)>0\\ (iv).f(x)=-1,& g(x)\text{dan}h(x)\\ & \text{keduanya ganjil}\\ & \text{atau genap}\end{array}\\ 7.& g(x{)}^{f(x)}=h(x{)}^{f(x)}& \{\begin{array}{ll}(i).g(x)=h(x)& \\ (ii).f(x)=0,& g(x)\ne 0,h(x)\ne 0\end{array}\\ 8.& \begin{array}{rl}& A{\left(a^{f(x)}\right)}^2\\ & +B\left(a^{f(x)}\right)\\ & +C=0\end{array}& a>0,a\ne 1\\ \hline\end{array}$

$\text{E.2 Pertidaksamaan Eksponen dan Logaritma}$

Berikut sifat pertidaksamaan Eksponen

$\begin{array}{|ll|}\hline a>1& 0<a<1\\ a^{f(x)}\le a^{g(x)}\Rightarrow f(x)\le g(x)& a^{f(x)}\le a^{g(x)}\Rightarrow f(x)\ge g(x)\\ a^{f(x)}<a^{g(x)}\Rightarrow f(x)<g(x)& a^{f(x)}<a^{g(x)}\Rightarrow f(x)>g(x)\\ a^{f(x)}\ge a^{g(x)}\Rightarrow f(x)\ge g(x)& a^{f(x)}\ge a^{g(x)}\Rightarrow f(x)\le g(x)\\ a^{f(x)}>a^{g(x)}\Rightarrow f(x)>g(x)& a^{f(x)}>a^{g(x)}\Rightarrow f(x)<g(x)\\ \hline\end{array}$

Untuk pertidaksamaan logaritma (dengan syarat  $(f(x)>0\text{dan}g(x)>0)$ ) adalah sebagai berikut:

$\begin{array}{|ll|}\hline a>1& 0<a<1\\ \begin{array}{rl}& {}^a\log f(x)\le {}^a\log g(x)\\ & \Rightarrow f(x)\le g(x)\end{array}& \begin{array}{rl}& {}^a\log f(x)\le {}^a\log g(x)\\ & \Rightarrow f(x)\ge g(x)\end{array}\\ \begin{array}{rl}& {}^a\log f(x)<{}^a\log g(x)\\ & \Rightarrow f(x)<g(x)\end{array}& \begin{array}{rl}& {}^a\log f(x)<{}^a\log g(x)\\ & \Rightarrow f(x)>g(x)\end{array}\\ \begin{array}{rl}& {}^a\log f(x)\ge {}^a\log g(x)\\ & \Rightarrow f(x)\ge g(x)\end{array}& \begin{array}{rl}& {}^a\log f(x)\ge {}^a\log g(x)\\ & \Rightarrow f(x)\le g(x)\end{array}\\ \begin{array}{rl}& {}^a\log f(x)>{}^a\log g(x)\\ & \Rightarrow f(x)>g(x)\end{array}& \begin{array}{rl}& {}^a\log f(x)>{}^a\log g(x)\\ & \Rightarrow f(x)<g(x)\end{array}\\ \hline\end{array}$