Expand the Logarithmic Expression log of log of 100000^(20x)

Problem

(log_)((log_)(100000(20*x)))

Solution

1. Identify the base of the logarithms. Since no base is specified, we assume the common logarithm with base 10, where (log_)(10)=n

2. Simplify the innermost expression by expressing 100,000 as a power of 10.

100000=10

3. Substitute this value into the power inside the inner logarithm.

100000(20*x)=(10)(20*x)

4. Apply the power rule for exponents, (am)n=a(m*n) to simplify the expression.

(10)(20*x)=10(100*x)

5. Evaluate the inner logarithm using the property (log_10)(10)=k

(log_)(10(100*x))=100*x

6. Substitute the result back into the outer logarithm.

(log_)(100*x)

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

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

8. Evaluate the constant logarithm (log_)(100) since 100=10

(log_)(100)=2

9. Combine the terms to reach the final expanded form.

2+(log_)(x)

Final Answer

(log_)((log_)(100000(20*x)))=2+(log_)(x)

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