If $x+\log _{15}\left(5+3^x\right)=x \log _{15} 5+\log _{15} 24$, then $x=\ldots .$.

Solution:

The equation $x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24$ is solved using logarithmic identities. Rewriting $x$ as $\log_{15}15^x$ and $x\log_{15}5$ as $\log_{15}5^x$ transforms the equation to:

$\log_{15}15^x+\log_{15}(5+3^x)=\log_{15}5^x+\log_{15}24$

Combining logarithms on each side gives:

$\log_{15}\left[15^x(5+3^x)\right]=\log_{15}(5^x\cdot24)$

Equating the arguments produces:

$15^x(5+3^x)=5^x\cdot24$

Dividing through by $5^x$ and simplifying yields $3^x(5+3^x)=24$. Substituting $y=3^x$ leads to the quadratic $y^2+5y-24=0$, which factors as $(y+8)(y-3)=0$.

The solution $y=-8$ is extraneous since $3^x>0$. The valid solution $y=3$ implies $3^x=3$, so $x=1$.

Verification confirms that $x=1$ satisfies the original equation.

Final answer: $\boxed{1}$