Question Video: Finding the Solution Set of Logarithmic Equations over the Set of Real Numbers Mathematics

Tue or False: The solution set of log₈ 32 − log₂ (𝑥 − 1) = 5/3 in ℝ is {2}.

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Video Transcript

Tue or False: The solution set of log base eight of 32 minus log base two of 𝑥 minus one equals five-thirds in the set of real values is the set containing two.

We will answer this question using the laws of logarithms. We begin by recalling that log base 𝑎 of 𝑏 can be rewritten as log base 𝑥 of 𝑏 divided by log base 𝑥 of 𝑎, where 𝑥 is positive and not equal to one. This means that we can rewrite the first term log base eight of 32 as log base 𝑥 of 32 divided by log base 𝑥 of eight. Noting that the base of our second term is two, together with the fact that two cubed equals eight and two to the fifth power equals 32, we will let the base 𝑥 equal two.

Next, we recall that if log base 𝑎 of 𝑏 equals 𝑦, then 𝑎 to the power of 𝑦 equals 𝑏. Using this together with the fact that two cubed equals eight and two to the fifth power is 32, then log base two of 32 is five and log base two of eight is three. The first term in our equation log base eight of 32 is therefore equal to five-thirds.

We will now clear some space and try and solve the full equation. We have five-thirds minus log base two of 𝑥 minus one is equal to five-thirds. Subtracting five-thirds and adding log base two of 𝑥 minus one to both sides, we have five-thirds minus five-thirds is equal to log base two of 𝑥 minus one. The left-hand side is equal to zero.

We can now use the fact that the log of one with any base equals zero to find the value of 𝑥. We need to solve the equation 𝑥 minus one equals one. Adding one to both sides gives us 𝑥 is equal to two. This means that the solution set to the equation log base eight of 32 minus log base two of 𝑥 minus one equals five-thirds is two. We can therefore conclude that the statement is true.

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