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

Solve the equation log_(𝑥) 𝑥³⁵ = 5𝑥, where 𝑥 ∈ ℝ.

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### Video Transcript

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

In order to answer this question, we will use the fact that if log base 𝑎 of 𝑏 is equal to 𝑐, then 𝑏 is equal to 𝑎 to the power of 𝑐. In this question, our values of 𝑎, 𝑏, and 𝑐 are 𝑥, 𝑥 to the power of 35, and five 𝑥, respectively. This means that we can rewrite the equation such that 𝑥 to the power of 35 is equal to 𝑥 to the power of five 𝑥.

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

The solution to the equation log base 𝑥 of 𝑥 to the power of 35 equals five 𝑥 is 𝑥 equals seven.