x +log₁₅^{(1+3^x)} = xlog₁₅⁵+ log₁₅¹² where x is an integer , then what is x equal to

• -3
• 2
• 1
• 3

To solve the equation $x+{\log }_{15}(1+3^x)=x{\log }_{15}5+{\log }_{15}12$, we need to simplify and analyze each part carefully. Our goal is to find integer values for $x$ that satisfy this equation, specifically checking the options -3, 2, 1, and 3.

Breaking Down the Equation

Start by rearranging the equation to isolate the logarithmic terms:

• Rewrite the equation as ${\log }_{15}(1+3^x)=x{\log }_{15}5+{\log }_{15}12-x$.

Next, we can use properties of logarithms to combine the right side:

• Recall that ${\log }_{15}a+{\log }_{15}b={\log }_{15}(a\cdot b)$.
• Thus, we can express the right side as ${\log }_{15}(12\cdot 5^x)-x$.

Evaluating the Logarithmic Equation

Now we have:

${\log }_{15}(1+3^x)={\log }_{15}(12\cdot 5^x)-x$

To eliminate the logarithm, we can exponentiate both sides, leading to:

$1+3^x=\frac{12\cdot 5^x}{{15}^x}$

Which simplifies to:

$1+3^x=\frac{12}{{15}^x}\cdot 5^x=\frac{12\cdot 5^x}{{15}^x}$

Testing Integer Values

Now let’s check the integer values for $x$ provided in your question: -3, 2, 1, and 3. We will evaluate the left-hand side and the right-hand side for each value.

1. When $x=-3$

Left side: $-3+{\log }_{15}(1+3^{-3})=-3+{\log }_{15}(1+\frac{1}{27})\approx -3+{\log }_{15}(\frac{28}{27})$

Right side: $-3{\log }_{15}5+{\log }_{15}12\approx -3\cdot 0.464+1.079$ which does not match.

2. When $x=2$

Left side: $2+{\log }_{15}(1+9)=2+{\log }_{15}(10)\approx 2+1$ which is around 3.

Right side: $2{\log }_{15}5+{\log }_{15}12\approx 2\cdot 0.464+1.079$ again does not match.

3. When $x=1$

Left side: $1+{\log }_{15}(1+3)=1+{\log }_{15}(4)\approx 1+0.654\approx 1.654.$

Right side: $1{\log }_{15}5+{\log }_{15}12\approx 0.464+1.079\approx 1.543$ does not match.

4. When $x=3$

Left side: $3+{\log }_{15}(1+27)=3+{\log }_{15}(28)\approx 3+1.072=4.072.$

Right side: $3{\log }_{15}5+{\log }_{15}12\approx 3\cdot 0.464+1.079\approx 1.392+1.079=2.471$ does not match.

Conclusion

After testing all the integer options provided, none of them satisfy the equation. If you were looking for a specific integer solution, it may be necessary to reevaluate the original equation or check for alternative integer values outside the ones given. Sometimes, equations can have no integer solutions, or they may require a different range of values to explore. If you have more questions or need further clarification, feel free to ask!