Question

50 points!!!!! what is the simplified form of 6 log3x - 7 log3y + 4 log3z? log base 3 of the quantity negative xz end quantity divided by 4y end log log base 3 of 3xz divided by y end log log base 3 of the quantity x to the sixth z to the fourth end quantity divided by y to the seventh end log log base 3 of the quantity xy all raised to the twelfth end quantity divided by z to the 4th end log

Answer 1

log base 3 of the quantity x to the seventh z to the fourth end quantity divided by y to the sixth end log

Sure hope this helps you :D

Answer 2

The simplified form of the logarithmic expression, 6 log₃x - 7 log₃y + 4 log₃z, is log₃{(x⁶z⁴)/y⁷}. Hence, the third option is the right choice.

What are the laws involved in the simplification of logarithmic expressions?

The laws involved in the simplification of logarithmic expressions are:

• Power law: $a log_x b= log _xb^a$
• Product law: $log_xa + log_xb = log_xab$
• Quotient law: $log_xa - log_xb = log_x\frac{a}{b}$
• Base changing law: $log_ab = \frac{log_xb}{log_xa}$

How to solve the question?

In the question, we are asked to simplify the logarithmic expression,

6 log₃x - 7 log₃y + 4 log₃z.

The simplification of the provided logarithmic expression can be done as follows:-

6 log₃x - 7 log₃y + 4 log₃z,

= log₃x⁶ - log₃y⁷ + log₃z⁴ {Using the Power law: $a log_x b= log _xb^a$}

= log₃x⁶ + log₃z⁴ - log₃y⁷

= log₃x⁶z⁴ - log₃y⁷ {Using the Product law: $log_xa + log_xb = log_xab$}

= log₃{(x⁶z⁴)/y⁷} {Using the Quotient law: $log_xa - log_xb = log_x\frac{a}{b}$}.

Thus, the simplified form of the logarithmic expression, 6 log₃x - 7 log₃y + 4 log₃z, is log₃{(x⁶z⁴)/y⁷}. Hence, the third option is the right choice.

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