Final answer:
The expression 2 log x + 3 log y - 5 log z can be simplified to log((x²y³)/z⁵) using properties of logarithms and exponents, making the correct answer option (a): log(x²) + log(y³) - log(z⁵).
Step-by-step explanation:
The student is asking how to express a combination of multiple logarithmic terms as a single logarithm. To do this, we need to use certain properties of logarithms and exponents.
Starting with the expression given, 2 log x + 3 log y - 5 log z, we can apply the third property mentioned, which tells us that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number:
- 2 log x = log(x²)
- 3 log y = log(y³)
- 5 log z = log(z⁵)
Then we apply the first property, which says the logarithm of a product of two numbers is the sum of the logarithms of the two numbers:
log(x²) + log(y³) = log(x²y³)
Finally, using the second property, which states that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, we can combine the remaining terms:
log(x²y³) - log(z⁵) = log((x²y³)/z⁵)
So, the expression 2 log x + 3 log y - 5 log z can be simplified as:
log((x²y³)/z⁵)
Comparing the answer choices provided, the correct answer is option (a): log(x²) + log(y³) - log(z⁵).