Question

The proof for the product property of logarithms requires simplifying the expression logb(bx y) to x y. Which property is used to justify this step?.

Answer 1

Answer:

logb(b^c)=c

Step-by-step explanation:

If you are doing this on edge it's C

Answer 2

You can use the properties of logarithm to derive the simplified form of the given expression.

The simplification of the given expression requires the given below properties of logarithm

• $log_a(b^c) = c \times log_a(b)$
• $log_b(b) = 1$

What is logarithm and some of its useful properties?

When you raise a number with an exponent, there comes a result.

Lets say you get

$a^b = c$

Then, you can write 'b' in terms of 'a' and 'c' using logarithm as follows

• $b = log_a(c)$

Some properties of logarithm are:

$log_a(b) = log_a(c) \implies b = c\\\\\log_a(b) + log_a(c) = log_a(b \times c)\\\\log_a(b) - log_a(c) = log_a(\frac{b}{c})\\\\log_a(b^c) = c \times log_a(b)\\\\log_b(b) = 1$

Using the above properties, to get to the simplified form of the given expression

The given expression is

$log_b(b^{x+y})$

Using the property $log_a(b^c) = c \times log_a(b)$, we get

$log_b(b^{x+y}) = (x+y)\times log_b(b)$

Using the property $log_b(b) = 1$, we get

$log_b(b^{x+y}) = (x+y)\times log_b(b) = (x+y) \times 1 = x + y$

Thus,

The simplification of the given expression requires the given below properties of logarithm

• $log_a(b^c) = c \times log_a(b)$
• $log_b(b) = 1$

Learn more about logarithms here:

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