Question

Let logbA=5;logbC=7;logbD=2 what is the value of logb(A^5C^2/D^6)

Answer 1

Answer:

A. 27

Step-by-step explanation:

$\text{De}\text{finition of logarithm}\ \log_ab=c\iff a^c=b.\\\\\log_bA=5\iff A=b^5\\\\\log_bC=7\iff C=b^7\\\\\log_bD=2\iff D=b^2\\\\\dfrac{A^5C^2}{D^6}=\dfrac{(b^5)^5(b^7)^2}{(b^2)^6}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\dfrac{b^{25}b^{14}}{b^{12}}\qquad\text{use}\ a^n\cdot a^m=a^{n+m}\ \text{and}\ \dfrac{a^m}{a^n}=a^{m-n}\\\\=b^{25+14-12}=b^{27}\\\Downarrow\\\\\log_b\left(\dfrac{A^5C^2}{D^6}\right)=\log_bb^{27}\qquad\text{use}\ \log_aa^n=n\\\\\log_b\left(\dfrac{A^5C^2}{D^6}\right)=27$