Use the properties of logarithms to evaluate $log _ { 27 } 9 + \log _ { 27 } 81$ without a calculator.

A) 2
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) 4
E) 1

Question

Use the properties of logarithms to evaluate   log ⁡ 27 9 + log ⁡ 27 81 \log _ { 27 } 9 + \log _ { 27 } 81 lo g 27 ​ 9 + lo g 27 ​ 81   without a calculator.A) 2B)    2 3  \frac{2}{3}  3 2 ​ C)    3 4  \frac{3}{4}  4 3 ​ D)  4E) 1

Miriam Rodriguez · Accepted Answer

Final Answer : A, Explanation :   The properties of logarithms allow us to simplify the expression. First, recognize that 9 = 27^(2/3) and 81 = 27^(4/3). Therefore,   log ⁡ 27 9 = 2 3 \log _ { 27 } 9 = \frac{2}{3} lo g 27 ​ 9 = 3 2 ​   and   log ⁡ 27 81 = 4 3 \log _ { 27 } 81 = \frac{4}{3} lo g 27 ​ 81 = 3 4 ​  . Adding these together gives   2 3 + 4 3 = 2 \frac{2}{3} + \frac{4}{3} = 2 3 2 ​ + 3 4 ​ = 2  .

Use the properties of logarithms to evaluate $log _ { 27 } 9 + \log _ { 27 } 81$ without a calculator.

A) 2
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) 4
E) 1

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Use the properties of logarithms to evaluate log⁡279+log⁡2781\log _ { 27 } 9 + \log _ { 27 } 81log27​9+log27​81 without a calculator.A) 2B) 23 \frac{2}{3} 32​ C) 34 \frac{3}{4} 43​ D) 4E) 1

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Learning Objectives

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Use the properties of logarithms to evaluate $log _ { 27 } 9 + \log _ { 27 } 81$ without a calculator.

A) 2
B) $\frac{2}{3}$
C) $\frac{3}{4}$
D) 4
E) 1