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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?.

Sagot :

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

  • [tex]log_a(b^c) = c \times log_a(b)\\\\[/tex]
  • [tex]log_b(b) = 1[/tex]

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

[tex]a^b = c[/tex]

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

  • [tex]b = log_a(c)[/tex]

Some properties of logarithm are:

[tex]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[/tex]

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

The given expression is

[tex]log_b(b^{x+y})[/tex]

Using the property [tex]log_a(b^c) = c \times log_a(b)\\\\[/tex], we get

[tex]log_b(b^{x+y}) = (x+y)\times log_b(b)[/tex]

Using the property [tex]log_b(b) = 1[/tex], we get

[tex]log_b(b^{x+y}) = (x+y)\times log_b(b) = (x+y) \times 1 = x + y[/tex]

Thus,

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

  • [tex]log_a(b^c) = c \times log_a(b)\\\\[/tex]
  • [tex]log_b(b) = 1[/tex]

Learn more about logarithms here:

https://brainly.com/question/20835449

Answer:

logb(b^c)=c

Step-by-step explanation:

If you are doing this on edge it's C

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