Final answer:
To solve the logarithmic equation log₇(3) - log₇(-5x) = 1, we apply the quotient rule of logarithms and convert the equation into an exponential form, which results in the solution x = -7/15.
Step-by-step explanation:
We are asked to solve a logarithmic equation given by log₇(3) - log₇(-5x) = 1. To solve for x, we will use properties of logarithms. According to the laws of logarithms, the equation can be rewritten using the quotient rule, which states that logb(A) - logb(B) = logb(A/B), where A and B are positive real numbers and b is the base of the logarithm.
Applying this property, we get log₇(3/(-5x)) = 1. To solve the equation, we need to eliminate the logarithmic term. We do this by converting the logarithmic equation into an exponential form: 7log₇(3/(-5x)) = 71, which simplifies to 3/(-5x) = 7. Multiplying both sides by -5x yields -15x = 7. Finally, we solve for x by dividing both sides by -15 to get x = -7/15.
Note that the original equation log₇(3) - log₇(-5x) = 1 assumes that x is such that -5x is positive, because logarithms are not defined for negative values. Therefore, the solution x = -7/15 is valid only if -5x > 0, which is true for x = -7/15 as this results in a positive quantity within the logarithm.