Video: Solving Equations Using Laws of Logarithms

Solve the equation ln 𝑒^(9π‘₯) = 3 ln 7, giving your answer to three decimal places.

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Video Transcript

Solve the equation natural log of 𝑒 to the nine π‘₯ power equals three times the natural log of seven, giving your answer to three decimal places.

The first thing that we can do is simplify the natural log of 𝑒. Natural log is the same thing as log base 𝑒, so if we have log base 𝑒 of 𝑒, we know that this is equal to one, because of the property log base π‘Ž of π‘Ž equals one, so log base 𝑒 of 𝑒 equals one, so one times nine π‘₯ is equal to three natural log of seven.

Next, let’s go ahead and bring the three that’s out front up with the exponents. So we have nine π‘₯ equals the natural log of seven cubed, and seven cubed is 343.

Now the natural log of 343 we can plug in our calculator. So we have nine π‘₯ equals 5.837730447. So now let’s divide both sides by nine. Therefore, π‘₯ is equal to 0.649, after rounding three decimal places.

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