Express in terms of sums and differences of logarithms:
log_a x^3 y^2 z
If log_a means log to base a, then that equals
3 log_a x + 2 log_a y + log_a z
To express log_a x^3 y^2 z in terms of sums and differences of logarithms, we can use the properties of logarithms.
First, let us express each term x^3, y^2, and z as separate logarithms:
log_a x^3 = 3 log_a x
log_a y^2 = 2 log_a y
log_a z = log_a z
Now, we can rewrite log_a x^3 y^2 z as follows:
log_a x^3 y^2 z = log_a x^3 + log_a y^2 + log_a z
Substituting the expressions we found earlier:
= 3 log_a x + 2 log_a y + log_a z
Therefore, log_a x^3 y^2 z can be expressed as the sum of 3 log_a x, 2 log_a y, and log_a z.
To express log_a(x^3 y^2 z) in terms of sums and differences of logarithms, we can use the logarithmic properties.
1. Start by using the power rule, which states that log_a(b^c) = c * log_a(b):
log_a(x^3 y^2 z) = 3 * log_a(x) + 2 * log_a(y) + log_a(z)
2. Next, apply the product rule, which states that log_a(b * c) = log_a(b) + log_a(c):
log_a(x^3 y^2 z) = 3 * (log_a(x) + log_a(y)) + log_a(z)
3. Finally, to express it as the sum and difference of logarithms, we can rewrite the above expression by separating the terms:
log_a(x^3 y^2 z) = 3 * log_a(x) + 3 * log_a(y) + log_a(z) - log_a(x) - log_a(y)
log_a(x^3 y^2 z) = 3 * log_a(x) - log_a(x) + 3 * log_a(y) - log_a(y) + log_a(z)
log_a(x^3 y^2 z) = 2 * log_a(x) + 2 * log_a(y) + log_a(z)
Therefore, log_a(x^3 y^2 z) expressed in terms of sums and differences of logarithms is 2 * log_a(x) + 2 * log_a(y) + log_a(z).