If log_5(9) = 1.365 and log_5(2)=0.431, then what is log_5(24)?

How to find the solution for this one?

Rules of logarithm

if log(a)=x, then log(a^2)=2x, ... log(a^n)=n×log(a)

Similarly,
if log(a)=x, then log(√a)=(1/2)x...

so, log 24 = (1/2) log9 + 3 log2

To find the value of log_5(24), we can use the properties of logarithms.

First, let's use the property of logarithms that states log_a(b) + log_a(c) = log_a(b * c). Using this property, we can rewrite log_5(24) as log_5(2 * 12), since 24 can be expressed as the product of 2 and 12.

Next, let's substitute the given values of log_5(2) and log_5(9) into the equation. log_5(2 * 12) becomes log_5(2) + log_5(12).

Now, we can use the property of logarithms that states log_a(b^c) = c * log_a(b). Applying this property, we can rewrite log_5(12) as log_5(2^2 * 3), since 12 can be expressed as the product of 2^2 and 3.

Using the same property again, log_5(12) becomes 2 * log_5(2) + log_5(3).

Substituting the given values, we have 2 * 0.431 + log_5(3).

Since we don't have the value of log_5(3) given, we can't simplify it further.

Therefore, the solution for log_5(24) is 2 * 0.431 + log_5(3), which cannot be determined without the value of log_5(3).