Solved on Dec 07, 2023

Simplify the expression: log10005\log \sqrt[5]{1000}.

STEP 1

Assumptions
1. We are working with logarithms to the base 10.
2. The expression inside the logarithm is a fifth root, which can be written as an exponent of 15\frac{1}{5}.
3. The number 1000 can be expressed as 10310^3.

STEP 2

Rewrite the fifth root of 1000 as an exponent.
10005=100015\sqrt[5]{1000} = 1000^{\frac{1}{5}}

STEP 3

Express 1000 as a power of 10.
1000=1031000 = 10^3

STEP 4

Combine the steps above to rewrite the expression inside the logarithm.
10005=(103)15\sqrt[5]{1000} = (10^3)^{\frac{1}{5}}

STEP 5

Use the power rule for exponents, which states that (am)n=amn(a^{m})^{n} = a^{mn}, to simplify the expression inside the logarithm.
(103)15=103×15(10^3)^{\frac{1}{5}} = 10^{3 \times \frac{1}{5}}

STEP 6

Multiply the exponents.
103×15=103510^{3 \times \frac{1}{5}} = 10^{\frac{3}{5}}

STEP 7

Now that we have the expression inside the logarithm simplified, we can apply the logarithm.
log10005=log1035\log \sqrt[5]{1000} = \log 10^{\frac{3}{5}}

STEP 8

Use the property of logarithms that states logab=bloga\log a^b = b \log a.
log1035=35log10\log 10^{\frac{3}{5}} = \frac{3}{5} \log 10

STEP 9

Since log10\log 10 is equal to 1 (because the base of the logarithm is 10), we can simplify further.
35log10=35×1\frac{3}{5} \log 10 = \frac{3}{5} \times 1

STEP 10

Calculate the final result.
35×1=35\frac{3}{5} \times 1 = \frac{3}{5}
The simplified expression is 35\frac{3}{5}.

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