Solved on Dec 07, 2023

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

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 $\frac{1}{5}$ .
3. The number 1000 can be expressed as $10^3$ .

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

Multiply the exponents.
$10^{3 \times \frac{1}{5}} = 10^{\frac{3}{5}}$

STEP 7

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

STEP 8

Use the property of logarithms that states $\log a^b = b \log a$ .
$\log 10^{\frac{3}{5}} = \frac{3}{5} \log 10$

STEP 9

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

STEP 10

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

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

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 $\frac{1}{5}$ .
3. The number 1000 can be expressed as $10^3$ .

STEP 2

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

STEP 3

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

STEP 4

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

STEP 5

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

STEP 6

Multiply the exponents.
$10^{3 \times \frac{1}{5}} = 10^{\frac{3}{5}}$

STEP 7

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

STEP 8

Use the property of logarithms that states $\log a^b = b \log a$ .
$\log 10^{\frac{3}{5}} = \frac{3}{5} \log 10$

STEP 9

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

STEP 10

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

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Simplify the expression: log⁡10005\log \sqrt[5]{1000}log51000​.

STEP 1

Assumptions1. 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}51​.3. The number 1000 can be expressed as 10310^3103.

STEP 2

Rewrite the fifth root of 1000 as an exponent.10005=100015\sqrt[5]{1000} = 1000^{\frac{1}{5}}51000​=100051​

STEP 3

Express 1000 as a power of 10.1000=1031000 = 10^31000=103

STEP 4

Combine the steps above to rewrite the expression inside the logarithm.10005=(103)15\sqrt[5]{1000} = (10^3)^{\frac{1}{5}}51000​=(103)51​

STEP 5

Use the power rule for exponents, which states that (am)n=amn(a^{m})^{n} = a^{mn}(am)n=amn, to simplify the expression inside the logarithm.(103)15=103×15(10^3)^{\frac{1}{5}} = 10^{3 \times \frac{1}{5}}(103)51​=103×51​

STEP 6

Multiply the exponents.103×15=103510^{3 \times \frac{1}{5}} = 10^{\frac{3}{5}}103×51​=1053​

STEP 7

Now that we have the expression inside the logarithm simplified, we can apply the logarithm.log⁡10005=log⁡1035\log \sqrt[5]{1000} = \log 10^{\frac{3}{5}}log51000​=log1053​

STEP 8

Use the property of logarithms that states log⁡ab=blog⁡a\log a^b = b \log alogab=bloga.log⁡1035=35log⁡10\log 10^{\frac{3}{5}} = \frac{3}{5} \log 10log1053​=53​log10

STEP 9

Since log⁡10\log 10log10 is equal to 1 (because the base of the logarithm is 10), we can simplify further.35log⁡10=35×1\frac{3}{5} \log 10 = \frac{3}{5} \times 153​log10=53​×1

STEP 10

Calculate the final result.35×1=35\frac{3}{5} \times 1 = \frac{3}{5}53​×1=53​The simplified expression is 35\frac{3}{5}53​.

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

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 $\frac{1}{5}$ .
3. The number 1000 can be expressed as $10^3$ .

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

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

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

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

Multiply the exponents.
$10^{3 \times \frac{1}{5}} = 10^{\frac{3}{5}}$

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

Use the property of logarithms that states $\log a^b = b \log a$ .
$\log 10^{\frac{3}{5}} = \frac{3}{5} \log 10$

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

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