Evaluate log of log of 100000^(200x)

Problem

(log_)((log_)(100000^(200*x)))

Solution

1. Interpret the expression as a common logarithm (base 10), where (log_)(a) represents (log_10)(a)

2. Simplify the innermost term by expressing the base as a power of 10.

(log_)((log_)((10^5)^(200*x)))

3. Apply the power rule for exponents, (a^m)^n=a^(m*n) to the inner term.

(log_)((log_)(10^(1000*x)))

4. Apply the logarithm property (log_b)(b^k)=k to the inner logarithm.

(log_)(1000*x)

5. Apply the product rule for logarithms, (log_)(a*b)=(log_)(a)+(log_)(b)

(log_)(1000)+(log_)(x)

6. Evaluate the constant logarithm (log_)(1000) since 10^3=1000

3+(log_)(x)

Final Answer

(log_)((log_)(100000^(200*x)))=3+(log_)(x)

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