# Question Video: Finding the Solution Set of Logarithmic Equations over the Set of Real Numbers Mathematics • 10th Grade

Solve the equation log_(π₯) π₯Β³β΅ = 5π₯, where π₯ β β.

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### Video Transcript

Solve the equation log base π₯ of π₯ to the power of 35 is equal to five π₯, where π₯ is in the set of all real numbers.

In order to answer this question, we will use the fact that if log base π of π is equal to π, then π is equal to π to the power of π. In this question, our values of π, π, and π are π₯, π₯ to the power of 35, and five π₯, respectively. This means that we can rewrite the equation such that π₯ to the power of 35 is equal to π₯ to the power of five π₯.

We notice here that the base of the left- and right-hand side are both π₯. As these are equal and π₯ is the base of the logarithm, which means it must be positive, then the exponents are also equal. 35 is equal to five π₯. We can divide both sides of this equation by five, leaving us with π₯ is equal to seven.

The solution to the equation log base π₯ of π₯ to the power of 35 equals five π₯ is π₯ equals seven.