Use the properties of logarithms to condense log ⁡ 12 ( x + 4 ) − 4 log ⁡ 12 x \log _ { 12 } ( x + 4 ) - 4 \log _ { 12 } x lo g 12 ( x + 4 ) − 4 lo g 12 x .A) log ⁡ 12 ( x + 4 − x 4 ) \log _ { 12 } \left( x + 4 - x ^ { 4 } \right) lo g 12 ( x + 4 − x 4 ) B) log ⁡ 12 ( x + 4 x ) 4 \log _ { 12 } \left( \frac { x + 4 } { x } \right) ^ { 4 } lo g 12 ( x x + 4 ) 4 C) log ⁡ 12 ( − 3 x + 4 ) \log _ { 12 } ( - 3 x + 4 ) lo g 12 ( − 3 x + 4 ) D) log ⁡ 12 x + 4 4 x \log _ { 12 } \frac { x + 4 } { 4 x } lo g 12 4 x x + 4 E) log ⁡ 12 x + 4 x 4 \log _ { 12 } \frac { x + 4 } { x ^ { 4 } } lo g 12 x 4 x + 4

Question

Use the properties of logarithms to condense   log ⁡ 12 ( x + 4 )  − 4 log ⁡ 12 x \log _ { 12 } ( x + 4 )  - 4 \log _ { 12 } x lo g 12 ​ ( x + 4 )  − 4 lo g 12 ​ x   .A)    log ⁡ 12 ( x + 4 − x 4 )  \log _ { 12 } \left( x + 4 - x ^ { 4 } \right)  lo g 12 ​ ( x + 4 − x 4 )  B)    log ⁡ 12 ( x + 4 x )  4 \log _ { 12 } \left( \frac { x + 4 } { x } \right)  ^ { 4 } lo g 12 ​ ( x x + 4 ​ )  4 C)    log ⁡ 12 ( − 3 x + 4 )  \log _ { 12 } ( - 3 x + 4 )  lo g 12 ​ ( − 3 x + 4 )  D)    log ⁡ 12 x + 4 4 x \log _ { 12 } \frac { x + 4 } { 4 x } lo g 12 ​ 4 x x + 4 ​ E)    log ⁡ 12 x + 4 x 4 \log _ { 12 } \frac { x + 4 } { x ^ { 4 } } lo g 12 ​ x 4 x + 4 ​

Regis Larkin · Accepted Answer

Final Answer : E, Explanation :   The properties of logarithms allow us to combine the terms into a single logarithm. The subtraction of logs corresponds to division, and the coefficient 4 in front of the second log can be rewritten as an exponent inside the log, leading to   log ⁡ 12 ( x + 4 ) − 4 log ⁡ 12 x = log ⁡ 12 x + 4 x 4 \log _ { 12 } ( x + 4 ) - 4 \log _ { 12 } x = \log _ { 12 } \frac { x + 4 } { x ^ { 4 } } lo g 12 ​ ( x + 4 ) − 4 lo g 12 ​ x = lo g 12 ​ x 4 x + 4 ​  .

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