Question

Condense the expression
3log₇ v + 6log₇ w - (log₇ u)/3

Answer 1

The expression 3log₇ v + 6log₇ w - (log₇ u)/3 can be condensed to log₇ ((v^3 * w^6) / u^(1/3)) using the laws of logarithms including the power, product, and quotient rules.

Step-by-step explanation:

To condense the logarithmic expression 3log₇ v + 6log₇ w - (log₇ u)/3, we can apply the laws of logarithms. Specifically, we use the power rule (alog(x) = log(x^a)), the product rule (log(x) + log(y) = log(xy)), and the quotient rule (log(x) - log(y) = log(x/y)).

1. Apply the power rule: Rewrite 3log₇ v as log₇ v^3 and 6log₇ w as log₇ w^6.
2. The term (log₇ u)/3 can also be rewritten using the power rule as log₇ u^(1/3), which means the cube root of u.
3. Apply the product rule: Combine log₇ v^3 and log₇ w^6 into log₇ (v^3 * w^6).
4. Apply the quotient rule: Combine log₇ (v^3 * w^6) and log₇ u^(1/3) to get log₇ ((v^3 * w^6) / u^(1/3)).

The expression is now condensed to a single logarithm: log₇ ((v^3 * w^6) / u^(1/3)).