If x+log _{15}left(5+3^xright)=x log _{15} 5+ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given the equation:$x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24$Rewrite $x$ as $\log_{15}(15^x)$:$\log_{15}(15^x)+\log_{15}(5+3^x)=\log_{15}(5^x)+\

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) Given the equation:$x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24$Rewrite $x$ as $\log_{15}(15^x)$:$\log_{15}(15^x)+\log_{15}(5+3^x)=\log_{15}(5^x)+\log_{15}24$Using the property $\log(a)+\log(b)=\log(ab)$:$\log_{15}(15^x(5+3^x))=\log_{15}(24 \cdot 5^x)$Equating the arguments:$15^x(5+3^x)=24 \cdot 5^x$Since $15^x = (3 \cdot 5)^x = 3^x \cdot 5^x$,we have:$3^x \cdot 5^x(5+3^x)=24 \cdot 5^x$Dividing both sides by $5^x$ (since $5^x 
eq 0$):$3^x(5+3^x)=24$Let $t=3^x$. Then:$t(5+t)=24$$t^2+5t-24=0$Factoring the quadratic equation:$(t+8)(t-3)=0$Since $t=3^x > 0$,we must have $t=3$:$3^x=3^1$$x=1$

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

Similar Questions

Let $x$ and $y$ be real numbers such that $x > 2y > 0$ and $2 \log (x - 2y) = \log x + \log y$. Then,the possible value$(s)$ of $\frac{x}{y}$ is/are:

$7\log \left( \frac{16}{15} \right) + 5\log \left( \frac{25}{24} \right) + 3\log \left( \frac{81}{80} \right) =$

The solution to the equation ${9^x} - {2^{x + 1/2}} = {2^{x + 3/2}} - {3^{2x - 1}}$ is:

$\log (\cosh 3 + \sinh 3) + \log (\cosh 3 - \sinh 3) = $

The number of solution$(s)$ of the equation $\ln(\ln x) = \log_x e$ is -

Vedclass Test Series

Exam Paper Generator

Online Exam Module