Final answer:
We used the properties of logarithms to find the value of various logarithmic expressions. The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the base number. In particular, we found log_7(49) to be 2, log_27(3) to be 1/3, log_2(1/128) to be -7, log(100) to be 2, and lne to be 1.
Step-by-step explanation:
The student's question involves solving logarithmic expressions and equations, which are a part of high school mathematics, particularly algebra and precalculus. We will go through the requested logarithmic values using the properties of logarithms.
Logarithmic Values:
- log7(49): Since 49 is 7 squared, we use the property that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the base number. Therefore, log7(72) = 2 * log7(7) = 2.
- log27(3): 27 is 3 cubed, so we can write this as log27(31/3) which equals 1/3.
- log2(1/128): Writing 128 as 27, we have log2(2-7) and thus this value is -7.
- log(100): Since 100 is 10 squared, log(100) = log(102) = 2.
- ln(e): The natural logarithm of e is 1, since e is the base of the natural logarithm. ln(e) = 1.
These calculations demonstrate the use of logarithmic properties including the power rule, the change of base, and the understanding that the natural logarithm of e is 1.