Final answer:
The exact value of the expression ln e⁵ · 32 log 3 (3) is 160. This result is found by recognizing that ln e⁵ equals 5 (since ln(e^x) = x), and 32 log 3 (3) simplifies to 32, because log base 3 of 3 is 1.
Step-by-step explanation:
The question involves finding the exact value of the expression ln e⁵ · 32 log 3 (3). The key to solving this problem is to use the properties of logarithms and exponential functions. The natural logarithm ln is the inverse of the exponential function with base e, where e is approximately 2.7182818. When taking the natural logarithm of an exponential function, like ln e⁵, the e and the ln cancel each other out, leaving you with just the exponent, which in this case is 5.
The second part, 32 log 3 (3), uses the logarithm with base 3. Since 3 raised to the power of what results in 3 is simply 1 (3ⁱ = 3), the value of log 3 (3) would be 1. Multiplying this by 32 gives us 32.
Putting it all together, we substitute the known values to get the final solution:
- ln e⁵ = 5
- 32 log 3 (3) = 32 * 1 = 32
- Now substitute the known quantity into the equation:
- 5 · 32 which equals 160.
Therefore, the exact value of the expression is 160.
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