Step 1: Understanding the Concept:
This is a logarithmic equation that involves an exponential function of \( x \).
Solving such equations requires using fundamental logarithm properties to simplify both sides into a single logarithm or a simple algebraic equality.
Specifically, we need to handle the standalone \( x \) and the coefficient \( x \) of the log term.
A key trick is converting any real number into a logarithm of a specific base using the definition \( y = \log_b b^y \).
Step 2: Key Formula or Approach:
Properties to use:
1. \( x = \log_{15} 15^x \).
2. \( n \log A = \log A^n \).
3. \( \log A + \log B = \log(AB) \).
Step 3: Detailed Explanation:
The equation is:
\[ x + \log_{15}(5 + 3^x) = x \log_{15} 5 + \log_{15} 24 \]
First, convert \( x \) and move the \( x \) coefficient:
\[ \log_{15} 15^x + \log_{15}(5 + 3^x) = \log_{15} 5^x + \log_{15} 24 \]
Apply the sum rule of logarithms to both sides:
\[ \log_{15} [15^x \cdot (5 + 3^x)] = \log_{15} [24 \cdot 5^x] \]
Since both sides have the same base, we can equate the arguments:
\[ 15^x (5 + 3^x) = 24 \cdot 5^x \]
Recall that \( 15^x = (5 \times 3)^x = 5^x \cdot 3^x \):
\[ (5^x \cdot 3^x)(5 + 3^x) = 24 \cdot 5^x \]
Since \( 5^x \) is always positive and non-zero for any real \( x \), we can divide both sides by \( 5^x \):
\[ 3^x (5 + 3^x) = 24 \]
Expanding the left side:
\[ (3^x)^2 + 5(3^x) - 24 = 0 \]
This is a quadratic equation in terms of \( 3^x \). Let \( t = 3^x \):
\[ t^2 + 5t - 24 = 0 \]
Factoring the quadratic:
\[ (t + 8)(t - 3) = 0 \]
The roots are \( t = -8 \) and \( t = 3 \).
Since \( t = 3^x \) and exponential functions always yield positive values, \( t = -8 \) is impossible.
Thus, \( 3^x = 3 \).
Comparing exponents, we get \( x = 1 \).
Step 4: Final Answer:
The solution is \( x = 1 \).
This is Option (B).