Can someone solve this showing the steps involved
4^log2(2^log2 5)
To solve the expression 4^log2(2^log2 5), let's break it down step by step.
Step 1: Simplify the innermost exponent, 2^log2 5.
We can use the property of logarithms that states log_b(b^x) = x if b > 0 and b ≠ 1.
In this case, b = 2 and x = log2 5.
So, 2^log2 5 simplifies to just 5.
Step 2: Substituting the simplified value into the original expression.
Now, we have 4^log2 5.
Step 3: Use the property of exponents, (a^b)^c = a^(b*c).
Here, a = 4, b = log2, and c = 5.
So, 4^log2 5 can be rewritten as (2^2)^log2 5.
Step 4: Apply the property of exponents (a^b)^c = a^(b*c).
(2^2)^log2 5 simplifies to 2^(2 * log2 5).
Step 5: Apply the property of logarithms, log_b(b^x) = x.
In this case, b = 2 and x = log2 5.
So, 2^(2 * log2 5) simplifies to 2^(log2 5^2).
Step 6: Simplify the exponent using the property of exponents a^(log_a x) = x.
Here, a = 2 and x = 5^2.
Therefore, 2^(log2 5^2) simplifies to just 5^2.
Step 7: Evaluate the final expression, 5^2.
5^2 equals 25.
Final Answer:
Therefore, 4^log2(2^log2 5) simplifies to 25.