Question

use the product rule of logarithms to write the completely expanded expression equivalent to log5(3x 6y). make sure to use parenthesis around your logarithm functions log(x y).

Answer 1

To expand the expression log5(3x · 6y) using the product rule of logarithms, we express it as the sum log5(3x) + log5(6y), which follows from the rule that log(xy) = log(x) + log(y).

Step-by-step explanation:

The product rule of logarithms states that the logarithm of a product of two numbers is equal to the sum of the logarithms of the individual numbers, which can be expressed as logb(xy) = logb(x) + logb(y). Similarly, when a number is raised to an exponent inside a logarithm, it can be brought out in front as a multiplier, expressed as logb(xy) = y · logb(x).

In the case of log5(3x · 6y), we can apply these rules by first seeing the expression as the product of two quantities (3x and 6y), and then expanding it to log5(3x) + log5(6y), which is the completely expanded expression equivalent to log5(3x · 6y) using the product rule of logarithms.

Answer 2

To expand the expression log5(3x^6 y), use the product rule to write it as log5(3) + log5(x^6) + log5(y), then apply the exponent rule to get the final expanded form: log5(3) + 6 · log5(x) + log5(y).

Step-by-step explanation:

To use the product rule of logarithms to write the completely expanded expression equivalent to log5(3x⁺ y⁺), you must remember that the logarithm of a product of two numbers is the sum of the logarithms of the individual numbers. The formula is log(xy) = log x + log y. Applying this rule and also using the rule that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number (log(xn) = n · log x), the expanded expression is:

log5(3) + log5(x⁺) + log5(y⁺)

The expression log5(x⁺) and log5(y⁺) can be further expanded by applying the exponent rule:

log5(3) + 6 · log5(x) + log5(y)