Answer :

Let's evaluate the expression step-by-step: [tex]5\log_2 \log_5 \log_{10} 10^{25}[/tex].

Step 1: Simplify the innermost logarithm, [tex]\log_{10} 10^{25}[/tex]:

The log base 10 of [tex]10^{25}[/tex] is simply the exponent because [tex]\log_{10} 10^x = x[/tex]. Hence, [tex]\log_{10} 10^{25} = 25[/tex].

Step 2: Move to the next logarithm, [tex]\log_5 25[/tex]:

We need to express 25 as a power of 5. Since [tex]25 = 5^2[/tex], we have [tex]\log_5 25 = \log_5 5^2 = 2\log_5 5[/tex].

Since [tex]\log_5 5 = 1[/tex] (because any log of its base is 1), [tex]\log_5 5^2 = 2\times1 = 2[/tex].

Step 3: Finally, evaluate [tex]5\log_2 2[/tex]:

Next, find [tex]\log_2 2[/tex]. We know [tex]\log_2 2 = 1[/tex] because the log of any number to its own base is 1.

Now, multiply by 5: [tex]5\times1 = 5[/tex].

So, the value of the expression [tex]5\log_2 \log_5 \log_{10} 10^{25}[/tex] simplifies to [tex]5[/tex].