Logarithm Question

Question

If  log₃(𝑥)=4,  log₃(𝑦)=−1, and  log₃(𝑧)=5 ,what is log₃[(xy)²/z³]?



Your answer should be an integer.

Answer 1

If  log₃(𝑥)=4,  log₃(𝑦)=−1, and  log₃(𝑧)=5 ,what is log₃[(xy)²/z³]?



First, convert logarithmic forms to exponential forms.

Rule: y = log_a(x) => x = a^y

log₃(x) = 4 => x = 3⁴ => x = 81

log₃(y) = -1 => y = 3^-1 => y = 1/3

log₃(z) = 5 => z = 3⁵ => z = 243

Next, substitute and simplify.

log₃[(xy)² / z³]

x = 3⁴

y = 3^-1

z = 3⁵

log₃[ (3⁴•3^-1)² / (3⁵)³ ]

log₃[ (3³)² / (3⁵)³ ]

log₃[ 3⁶ / 3¹⁵ ]

log₃[ 3^-9 ]

Now the exponent will come in front of the log and tada! We have our answer :)

-9 • log₃(3)

-9 • 1

-9 ✓

Finally,

If  log₃(𝑥)=4,  log₃(𝑦)=−1, and  log₃(𝑧)=5 ,

Then log₃[(xy)²/z³] -9.

Answer 2

log₃[(xy)²/z³] = log₃(xy)² - log₃(z³) = 2log₃(xy) - 3log₃(z) = 2[log₃(x) + log₃(y)] - 3log₃(z) =

2log₃(x) + 2log₃(y) - 3log₃(z)

You can plug in the values of each from what is given.