Evaluate log of fifth root of 10^9

Problem

(log_)(√(5,10))

Solution

1. Rewrite the expression using fractional exponents. The fifth root of a number is equivalent to raising that number to the power of 1/5

(log_)((10)1/5)

2. Apply the power of a power rule for exponents, which states that (am)n=a(m⋅n)

(log_)(10)

3. Use the power property of logarithms, (log_b)(xp)=p*(log_b)(x) to move the exponent in front of the logarithm.

9/5*(log_)(10)

4. Evaluate the common logarithm. Since (log_)() without a specified base denotes base 10 and (log_10)(10)=1 the expression simplifies further.

9/5⋅1

5. Simplify the final value into decimal form if necessary.

1.8

Final Answer

(log_)(√(5,10))=1.8

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