Question

Si $\log_{10}2=a$ y $\log_{10}3=b,$ entonces $\log_5 12$ es igual a:

If $\log_{10}2=a$ and $\log_{10}3=b,$ then $\log_5 12$ equals:

$\dfrac{a+b}{a+1}$

$\dfrac{2a+b}{a+1}$

$\dfrac{a+2b}{1+a}$

$\dfrac{2a+b}{1-a}$

$\dfrac{a+2b}{1-a}$

Usa la fórmula de cambio de base con logaritmos comunes

Use the change-of-base formula with common logarithms

Escribe $12=2^2\cdot3$ y $5=\frac{10}{2}$

Write $12=2^2\cdot3$ and $5=\frac{10}{2}$

Por cambio de base, $\begin{aligned} \log_5 12 &=\frac{\log12}{\log5}\\ &=\frac{2\log2+\log3} {\log10-\log2}\\ &=\frac{2a+b}{1-a}. \end{aligned}$

Por lo tanto, la respuesta correcta es D.

By change of base, $\begin{aligned} \log_5 12 &=\frac{\log12}{\log5}\\ &=\frac{2\log2+\log3} {\log10-\log2}\\ &=\frac{2a+b}{1-a}. \end{aligned}$

Therefore, the correct answer is D.