23. What is the solution of 5^(x-2)+3=32 ? A. 4.085 B. 4.092 C. 4.096 D. 4.099 E. 4.102

1. Start with the equation:
   \[5^ { ( x- 2) } + 3 = 32\]
2. Subtract 3 from both sides:
   \[5^ { ( x- 2) } = 29\]
3. Take the logarithm of both sides (using the natural logarithm, \(\ln \)):
   \[\ln ( 5^ { ( x- 2) } ) = \ln ( 29) \]
4. Use the logarithm power rule:
   \[( x- 2) \ln ( 5) = \ln ( 29) \]
5. Solve for \(x\):
   \[x- 2 = \frac { \ln ( 29) } { \ln ( 5) } \]
   \[x = 2 + \frac { \ln ( 29) } { \ln ( 5) } \]
6. Calculate the value using a calculator:
   \[x = 2 + \frac { \ln ( 29) } { \ln ( 5) } \approx 2 + 2.092 = 4.092\]
   Therefore, the solution is \(x \approx 4.092\).

Supplemental Knowledge

Exponential equations are defined as equations where variables appear as exponents. To solve them, logarithms - which act like their opposite in terms of exponential functions - often come into play; mastery of manipulating and solving exponential equations can have important ramifications across science, engineering and finance fields.

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