Evaluate log of log of 100^(50x)

Problem

(log_)((log_)(100(50*x)))

Solution

1. Identify the base of the logarithm. Since no base is specified, we assume the common logarithm with base 10 denoted as (log_10)() or simply (log_)()

2. Apply the power rule for logarithms, (log_)(ab)=b*(log_)(a) to the inner expression.

(log_)(100(50*x))=50*x*(log_)(100)

3. Evaluate the constant logarithm. Since 100=10 we know that (log_)(100)=2

50*x*(log_)(100)=50*x(2)

4. Simplify the inner expression by performing the multiplication.

50*x(2)=100*x

5. Substitute the simplified inner expression back into the outer logarithm.

(log_)(100*x)

6. Apply the product rule for logarithms, (log_)(a*b)=(log_)(a)+(log_)(b) to expand the expression.

(log_)(100*x)=(log_)(100)+(log_)(x)

7. Evaluate the remaining constant logarithm, (log_)(100)=2

2+(log_)(x)

Final Answer

(log_)((log_)(100(50*x)))=2+(log_)(x)

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