Solve for \(x\):
\[
x+\log_{15}(5+3^x)=x\log_{15}5+\log_{15}24
\]
Show Hint
Whenever a variable appears outside logarithms, convert it into logarithmic form using:
\[
x=\log_b(b^x)
\]
This helps in combining all logarithmic expressions into a single equation.
Step 1: Understanding the Concept:
This problem repeats Question 6. It requires solving an equation combining linear and logarithmic functions.
The core strategy is unifying all terms into a common logarithmic base to solve for the exponent. Step 2: Key Formula or Approach:
Use \( \log_a(mn) = \log_a m + \log_a n \) and \( \log_a a^k = k \). Step 3: Detailed Explanation:
Rewriting \( x \) as \( \log_{15} 15^x \):
\[ \log_{15} 15^x + \log_{15}(5+3^x) = \log_{15} 5^x + \log_{15} 24 \]
Combining terms:
\[ \log_{15} [15^x(5+3^x)] = \log_{15} (24 \cdot 5^x) \]
Equating arguments:
\[ 15^x(5+3^x) = 24 \cdot 5^x \]
Dividing by \( 5^x \):
\[ 3^x(5+3^x) = 24 \]
Solving for \( 3^x \):
\[ (3^x)^2 + 5(3^x) - 24 = 0 \]
Factors are \( 8 \) and \( -3 \), but for \( +5t \), they are \( (t+8)(t-3) = 0 \).
\( 3^x = 3 \implies x = 1 \). Step 4: Final Answer:
The answer is 1.
This matches Option (B).
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