But How Are They Connected?

Relevant wiki: Properties of Logarithms - Basic

$3^a=15\Rightarrow 3=15^{\frac{1}{a}}$

$5^b=15\Rightarrow 5=15^{\frac{1}{b}}$

$3\times 5=15^{\frac{1}{a}+\frac{1}{b}}=15 \therefore \boxed{\frac{1}{a}+\frac{1}{b}=1}.$

$a=\log_3 15$ , $b=\log_5 15$ .

$\dfrac 1 a + \dfrac 1 b = \log_{15} 3+ \log_{15} 5 = \log_{15} 15 = \boxed1$ .

$3^a=15$ $\implies$ $\ln 3^a=\ln 15$ $\implies$ $a=\dfrac{\ln 15}{\ln 3}$ $\implies$ $\dfrac{1}{a}=\dfrac{\ln 3}{\ln 15}$

$5^b=15$ $\implies$ $\ln 5^b=\ln 15$ $\implies$ $b=\dfrac{\ln 15}{\ln 5}$ $\implies$ $\dfrac{1}{b}=\dfrac{\ln 5}{\ln 15}$

$\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{\ln 3}{\ln 15} + \dfrac{\ln 5}{\ln 15} = 1.00$

$3^{a}=15$

$3^{a-1}=5$

$(3^{a-1})^b=15$

$\implies a=b(a-1)$

$a + b = ab$

${{a + b}\over{ab}}=1$

${1\over a}+{1\over b}=1$