Property 13
\(a\in \mathbb{R}\setminus\{0,1,-1\}\)
\(b\in \mathbb{R}\setminus\{0,1,-1\}\)
\(\left.\begin{aligned}a^x=b^y\\a^m=b^n\end{aligned}\right\}\)
\(\Rightarrow\samefrac xm=\samefrac yn\)
1.
\(2^x=3^4\)
\(2^6=3^y\)
\(\Rightarrow x\cdot y=?\)
2.
\(5^x=6^4\)
\(6^{6y}=5^{15}\)
\(\Rightarrow x\cdot y=?\)
3.
\(4^x=3^6\)
\(8^y=3^4\)
\(\Rightarrow\samefrac xy=?\)
4.
\(25^x=7^8\)
\(5^y=7^4\)
\(\Rightarrow\samefrac xy=?\)
5.
\(125^x=64\)
\(8^y=25\)
\(\Rightarrow x\cdot y=?\)
6.
\(81=125^x\)
\(5^y=3\)
\(\Rightarrow\samefrac xy=?\)
Property 14
Ordering in Exponential Expressions
- \(a>1\)
\(a^x<a^y\Rightarrow x<y\)
- \(0<a<1\)
\(a^x<a^y\Rightarrow x>y\)
Ordering of the expressions below.
1.
\(a=2^{-3}\)
\(b=2^{-1}\)
\(c=2^{-7}\)
2.
\(x=\left(\samefrac{1}{3}\right)^{10}\)
\(y=\left(\samefrac{1}{3}\right)^{12}\)
\(z=\left(\samefrac{1}{3}\right)^{20}\)
3.
\(x=(-2^4)^3\)
\(y=(-2^3)^4\)
\(z=-2^{(4^3)}\)
4.
\(x=\left(\samefrac{1}{2}\right)^{\samefrac{1}{3}}\)
\(y=\left(\samefrac{1}{2}\right)^{\samefrac{1}{5}}\)
\(z=\left(\samefrac{1}{2}\right)^{\samefrac{1}{7}}\)
5.
\(a=5^{28}\)
\(b=3^{42}\)
\(c=2^{56}\)