help with expanding
log base 5(x^2 times y)^4
log5 (x^2y)^4
4 log5(x^2y)
4 [ log5 (x^2)+ log ( y) ]
4 [ 2 log5 (x)+ log ( y) ]
8 log5(x) + 4 log5(y)
To expand the expression log base 5 of (x^2 * y)^4, we can use the properties of logarithms. Here's how:
Step 1: Apply the power rule of logarithms
The power rule states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of the number. In this case, we have (x^2 * y)^4 as the argument of the logarithm, so we can rewrite it as follows:
log base 5 of (x^2 * y)^4 = 4 * log base 5 of (x^2 * y)
Step 2: Apply the product rule of logarithms
The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In this case, we have the product x^2 * y, so we can rewrite the expression further:
4 * log base 5 of (x^2 * y) = 4 * (log base 5 of x^2 + log base 5 of y)
Step 3: Simplify the logarithms of the individual factors
The logarithm of x^2 to the base 5 can be simplified to 2 * log base 5 of x using the power rule. Similarly, the logarithm of y to the base 5 is log base 5 of y. Substituting these simplifications into the previous expression, we get:
4 * (2 * log base 5 of x + log base 5 of y)
So, the expanded form of log base 5 of (x^2 * y)^4 is 4 * (2 * log base 5 of x + log base 5 of y).