Question

Log3 5 * log5 8/ log3 4

Answer 1

= Loq3 8 /Loq3 .4 =Loq4 8 = 3/2 .

Answer 2

To simplify the expression:

log3(5) * log5(8) / log3(4)

We can use logarithm properties, specifically the change of base formula, which states:

log_b(a) = log_c(a) / log_c(b)

Let's apply this formula to each term:

1. log3(5) = log(5) / log(3)
2. log5(8) = log(8) / log(5)
3. log3(4) = log(4) / log(3)

Now, the expression becomes:

(log(5) / log(3)) * (log(8) / log(5)) / (log(4) / log(3))

Next, we can simplify further by canceling out common logarithms:

(log(5) / log(3)) * (log(8) / log(5)) / (log(4) / log(3)) = (log(5) / log(5)) * (log(8) / log(4))

Since log(a) / log(a) = 1 for any positive number a, we have:

1 * (log(8) / log(4))

Now, we can simplify the remaining logarithms:

log(8) = log(2^3) = 3 * log(2) log(4) = log(2^2) = 2 * log(2)

So, the expression becomes:

1 * (3 * log(2) / (2 * log(2)))

Now, we can cancel out the common factor of log(2):

(3/2) * (log(2) / log(2)) = 3/2

Therefore, the simplified value of the expression log3(5) * log5(8) / log3(4) is 3/2.