B3. Solve the equation: $1+\log \left(2^{x}+1\right)=\log 2+\log \left(4^{x}+9\right)$.

## 21st Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools National Competition, May 15, 2021

## Problems for 4th Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Enter your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-14.jpg?height=168&width=804&top_left_y=590&top_left_x=638)
B3. By applying the rule for the sum of logarithms, we transform the given equation into $\log 10\left(2^{x}+1\right)=$ $\log 2\left(4^{x}+9\right)$. We antilogarithmize the equation to the form $10\left(2^{x}+1\right)=2\left(4^{x}+9\right)$ and, by introducing a new unknown $2^{x}=t$, transform it into the equation $t^{2}-5 t+4=0$. The solutions to the equation are $t_{1}=1$ and $t_{2}=4$. We then calculate the values of the original unknowns of the equation: $x_{1}=0$ and $x_{2}=2$.

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=68&width=1757&top_left_y=2242&top_left_x=158)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=57&width=1753&top_left_y=2302&top_left_x=160)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=65&width=1753&top_left_y=2349&top_left_x=160)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=54&width=1753&top_left_y=2400&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=63&width=1756&top_left_y=2444&top_left_x=161)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=60&width=1753&top_left_y=2494&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=57&width=1753&top_left_y=2553&top_left_x=163)

# dUFA 

21st competition in mathematics for students of secondary technical and vocational schools National competition, May 15, 2021 Solutions to problems for 4th year

$$
\begin{array}{|c|c|c|}
\hline \mathrm{A} 1 & \mathrm{~A} 2 & \mathrm{~A} 3 \\
\hline \mathrm{E} & \mathrm{C} & \mathrm{C} \\
\hline
\end{array}
$$