Given $$\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$$ find the logarithm without using a calculator.
-$$\log _ { 10 } \sqrt { 24 }$$
A) 0.6901
B) 1.0792
C) 0.5396
D) 1.1748

Question

Quizplus · Accepted Answer

First, express $\sqrt{24}$ in terms of its prime factors: $\sqrt{24} = \sqrt{2^3 \cdot 3} = 2^{3/2} \cdot 3^{1/2}$. Then, use the given logarithm values and properties of logarithms to find $\log_{10} \sqrt{24}$: $\log_{10} 2^{3/2} + \log_{10} 3^{1/2} = \frac{3}{2} \cdot 0.3010 + \frac{1}{2} \cdot 0.4771 = 0.4515 + 0.2386 = 0.6901$.

Given log⁡102=0.3010 and log⁡103=0.4771, \log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }log10​2=0.3010 and log10​3=0.4771,

Given $\log _ { 10 } 2 = 0.3010 \text { and } \log _ { 10 } 3 = 0.4771 \text {, }$