Question

Log9 √5 * log25 √3

Answer 1

$log_{9} \sqrt{5}*log _{25} \sqrt{3}=log _{9} 5^{\frac{1}{2}}*log _{25} 3^{\frac{1}{2}}= \frac{1}{4}*log _{9}5*log _{25}3= \frac{1}{4}*log _{3} 5^{\frac{1}{2}}*log _{5} 3^{\frac{1}{2}}= \frac{1}{16}log _{3}5*log _{5}3= \frac{1}{16}*1= \frac{1}{16}$

Answer 2

$\log_{9}\sqrt{5}\times\log_{25}\sqrt{3}=\log_{3^{2}}5^{\frac{1}{2}}\times\log_{5^{2}}3^{\frac{1}{2}}=\frac{1}{4}\times\log_{3}5\times\frac{1}{4}\times\log_{5}3=\frac{1}{4}\times\frac{1}{4}\times1=\frac{1}{16}=0.0625=2^{-4}$

Answer 3

To simplify the expression, we can use logarithm properties to combine the two logarithms:

1. log_b(xy) = log_b(x) + log_b(y)
2. log_b(√x) = (1/2) * log_b(x)

Using these properties, let's simplify the expression:

log9(√5) * log25(√3)

Step 1: Simplify each logarithm separately log9(√5) = (1/2) * log9(5) log25(√3) = (1/2) * log25(3)

Step 2: Convert the bases to a common base. We can use the change of base formula: log_b(a) = log_c(a) / log_c(b)

Let's choose log2 as the common base:

log9(√5) = (1/2) * (log2(5) / log2(9)) log25(√3) = (1/2) * (log2(3) / log2(25))

Step 3: Simplify further:

log9(√5) = (1/2) * (log2(5) / log2(9)) = (1/2) * (log2(5) / (log2(3^2))) = (1/2) * (log2(5) / (2 * log2(3))) = (1/2) * (1/2) * (log2(5) / log2(3))

log25(√3) = (1/2) * (log2(3) / log2(25)) = (1/2) * (log2(3) / (log2(5^2))) = (1/2) * (log2(3) / (2 * log2(5))) = (1/2) * (1/2) * (log2(3) / log2(5))

Step 4: Combine the expressions: log9(√5) * log25(√3) = (1/2) * (1/2) * (log2(5) / log2(3)) * (1/2) * (1/2) * (log2(3) / log2(5))

The common factors cancel out:

log9(√5) * log25(√3) = (1/2) * (1/2) * (1/2) * (1/2) = 1/16

So, the final simplified value of the expression is 1/16.