Use $\log _ { 36 } 6 = 0.5000$ , $\log _ { 36 } 4 \approx 0.3869$ , and the properties of logarithms to approximate $\log _ { 36 } 144$ to four decimal places. Do not use a calculator.
A) $4.9678$
B) $4.5967$
C) $4.4506$
D) $3.4434$
E) $1.3869$

Question

Quizplus · Accepted Answer

$\log_{36}144$ can be simplified using the property of logarithms that $\log_b(a^n) = n \cdot \log_b(a)$. Since $144 = 36^2$, $\log_{36}144 = \log_{36}(36^2) = 2\cdot\log_{36}36 = 2\cdot1 = 2$. However, none of the options match this directly. The given values can be used to approximate $\log_{36}144$ by expressing 144 as a product of 6 and 4, both of whose logarithms are given. Specifically, $144 = 6^2 \cdot 4$, so $\log_{36}144 = \log_{36}(6^2) + \log_{36}(4) = 2\cdot\log_{36}(6) + \log_{36}(4) = 2\cdot0.5000 + 0.3869 = 1.3869$.

Use log⁡366=0.5000\log _ { 36 } 6 = 0.5000log36​6=0.5000 log⁡364≈0.3869\log _ { 36 } 4 \approx 0.3869log36​4≈0.3869

Use $\log _ { 36 } 6 = 0.5000$ $\log _ { 36 } 4 \approx 0.3869$