Property 7
\(\blacksquare\quad a^{\log_a b}=b\)
\(\blacksquare\quad a^{\log_b c}=c^{\log_b a}\)
\(a\longleftrightarrow c\)
1.
\(2^{\log_2 5}=?\)
2.
\(9^{\log_3 2}=?\)
3.
\(2^{\log_2 25}-3^{\log_9 16}=?\)
4.
\(5^{\log_5[\log_3 9]}=?\)
5.
\(e^{1+\ln(2x+2)}=6\cdot e\)
\(\Rightarrow x=?\)
6.
\(2^{\log x}+x^{\log2}=8\)
\(\Rightarrow x=?\)
7.
\(2^{\ln x}+3\cdot x^{\ln2}-8=0\)
\(\Rightarrow x=?\)
Property 8
\[f(x)=\log_{g(x)}h(x)\]
While the widest set of definition of a logarithmic function is found following properties should be considered.
- \(g(x)>0\)
- \(h(x)>0\)
- \(g(x)\ne1\)
What is the widest set of definition in following functions?
1.
\(f(x)=\log_2(x-3)\)
2.
\(f(x)=\log_4(5-x)\)
3.
\(f(x)=\log_{(x-4)}7\)
4.
\(f(x)=\log_x9\)
5.
\(f(x)=\log_{(x-7)}(20-x)\)
6.
\(f(x)=\log_{(x^2)}(x+3)\)