Math / Algebra

QuestionWrite the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. $\log \left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right)=$

Studdy Solution

STEP 1

1. We are given a logarithmic expression and need to express it as a sum or difference of logarithms.
2. We will use logarithmic properties to simplify the expression.
3. The expression involves a division and a square root, which can be simplified using logarithmic identities.

STEP 2

1. Apply the quotient rule for logarithms.
2. Simplify the logarithm of the numerator.
3. Simplify the logarithm of the denominator using the power rule.

STEP 3

Apply the quotient rule for logarithms, which states that:
$\log\left(\frac{M}{N}\right) = \log(M) - \log(N)$
For our expression:
$\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = \log(1000) - \log(\sqrt{a^{2}+b^{2}})$

STEP 4

Simplify the logarithm of the numerator, $\log(1000)$ .
Since $1000 = 10^3$ , we have:
$\log(1000) = \log(10^3) = 3 \log(10)$
And since $\log(10) = 1$ , it simplifies to:
$3 \log(10) = 3$

STEP 5

Simplify the logarithm of the denominator using the power rule, which states:
$\log(\sqrt{a^{2}+b^{2}}) = \log((a^{2}+b^{2})^{1/2}) = \frac{1}{2} \log(a^{2}+b^{2})$

STEP 6

Combine the simplified terms from Steps 2 and 3:
$\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = 3 - \frac{1}{2} \log(a^{2}+b^{2})$
This is the expression written as a sum or difference of logarithms, fully simplified.

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QuestionWrite the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. $\log \left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right)=$

Studdy Solution

STEP 1

1. We are given a logarithmic expression and need to express it as a sum or difference of logarithms.
2. We will use logarithmic properties to simplify the expression.
3. The expression involves a division and a square root, which can be simplified using logarithmic identities.

STEP 2

1. Apply the quotient rule for logarithms.
2. Simplify the logarithm of the numerator.
3. Simplify the logarithm of the denominator using the power rule.

STEP 3

Apply the quotient rule for logarithms, which states that:
$\log\left(\frac{M}{N}\right) = \log(M) - \log(N)$
For our expression:
$\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = \log(1000) - \log(\sqrt{a^{2}+b^{2}})$

STEP 4

Simplify the logarithm of the numerator, $\log(1000)$ .
Since $1000 = 10^3$ , we have:
$\log(1000) = \log(10^3) = 3 \log(10)$
And since $\log(10) = 1$ , it simplifies to:
$3 \log(10) = 3$

STEP 5

Simplify the logarithm of the denominator using the power rule, which states:
$\log(\sqrt{a^{2}+b^{2}}) = \log((a^{2}+b^{2})^{1/2}) = \frac{1}{2} \log(a^{2}+b^{2})$

STEP 6

Combine the simplified terms from Steps 2 and 3:
$\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = 3 - \frac{1}{2} \log(a^{2}+b^{2})$
This is the expression written as a sum or difference of logarithms, fully simplified.

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QuestionWrite the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. log⁡(1000a2+b2)=\log \left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right)=log(a2+b2​1000​)=

Studdy Solution

STEP 1

1. We are given a logarithmic expression and need to express it as a sum or difference of logarithms.2. We will use logarithmic properties to simplify the expression.3. The expression involves a division and a square root, which can be simplified using logarithmic identities.

STEP 2

1. Apply the quotient rule for logarithms.2. Simplify the logarithm of the numerator.3. Simplify the logarithm of the denominator using the power rule.

STEP 3

STEP 4

STEP 5

Simplify the logarithm of the denominator using the power rule, which states:log⁡(a2+b2)=log⁡((a2+b2)1/2)=12log⁡(a2+b2)\log(\sqrt{a^{2}+b^{2}}) = \log((a^{2}+b^{2})^{1/2}) = \frac{1}{2} \log(a^{2}+b^{2})log(a2+b2​)=log((a2+b2)1/2)=21​log(a2+b2)

STEP 6

Combine the simplified terms from Steps 2 and 3:log⁡(1000a2+b2)=3−12log⁡(a2+b2)\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = 3 - \frac{1}{2} \log(a^{2}+b^{2})log(a2+b2​1000​)=3−21​log(a2+b2)This is the expression written as a sum or difference of logarithms, fully simplified.

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QuestionWrite the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. $\log \left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right)=$

1. We are given a logarithmic expression and need to express it as a sum or difference of logarithms.
2. We will use logarithmic properties to simplify the expression.
3. The expression involves a division and a square root, which can be simplified using logarithmic identities.

1. Apply the quotient rule for logarithms.
2. Simplify the logarithm of the numerator.
3. Simplify the logarithm of the denominator using the power rule.

Simplify the logarithm of the denominator using the power rule, which states:
$\log(\sqrt{a^{2}+b^{2}}) = \log((a^{2}+b^{2})^{1/2}) = \frac{1}{2} \log(a^{2}+b^{2})$

Combine the simplified terms from Steps 2 and 3:
$\log\left(\frac{1000}{\sqrt{a^{2}+b^{2}}}\right) = 3 - \frac{1}{2} \log(a^{2}+b^{2})$
This is the expression written as a sum or difference of logarithms, fully simplified.