It is given that log x = 6 and log y = −2. Evaluate (a) log (xy) (b) log (1/sqrt(x))

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Question 1:
Given $\log x = 6$ and $\log y = -2$ .

a) Evaluate $\log(xy)$ .

Use the logarithm property $\log(AB) = \log A + \log B$ .
$\log(xy) = \log x + \log y$
Substitute the given values.
$\log(xy) = 6 + (-2) = 4$
The value is $\boxed{\text{4}}$ .

b) Evaluate $\log\left(\frac{1}{\sqrt{x}}\right)$ .
Rewrite the expression using exponent rules.
$\log\left(\frac{1}{\sqrt{x}}\right) = \log(x^{-1/2})$
Use the logarithm property $\log(A^B) = B \log A$ .
$\log(x^{-1/2}) = -\frac{1}{2} \log x$
Substitute the given value for $\log x$ .
$-\frac{1}{2} \log x = -\frac{1}{2} \times 6 = -3$
The value is $\boxed{\text{-3}}$ .

Question 2:
Evaluate:

a) $\log 16 \div \log 4$ .
Rewrite 16 as a power of 4.
$\frac{\log 16}{\log 4} = \frac{\log(4^2)}{\log 4}$
Use the logarithm property $\log(A^B) = B \log A$ .
$\frac{2 \log 4}{\log 4}$
Simplify the expression.
$\frac{2 \log 4}{\log 4} = 2$
The value is $\boxed{\text{2}}$ .

b) $\log 5 - \log 0.5$ .
Use the logarithm property $\log A - \log B = \log\left(\frac{A}{B}\right)$ .
$\log 5 - \log 0.5 = \log\left(\frac{5}{0.5}\right)$
Simplify the fraction inside the logarithm.
$\log\left(\frac{5}{0.5}\right) = \log(10)$
Evaluate $\log 10$ (assuming base 10).
$\log(10) = 1$
The value is $\boxed{\text{1}}$ .

Question 3:
Given $\log 6 = 0.7781$ and $\log 5 = 0.6990$ .

a) Calculate $\log 30$ .
Rewrite 30 as a product of 5 and 6.
$\log 30 = \log(5 \times 6)$
Use the logarithm property $\log(AB) = \log A + \log B$ .
$\log(5 \times 6) = \log 5 + \log 6$
Substitute the given values and calculate.
$\log 5 + \log 6 = 0.6990 + 0.7781 = 1.4771$
The value is $\boxed{\text{1.4771}}$ .

b) Calculate $\log 1\,200\,000$ .
Express $1\,200\,000$ in terms of 10 and a number that can be related to 5 and 6.
$\log 1\,200\,000 = \log(12 \times 10^5)$
Use the logarithm property $\log(AB) = \log A + \log B$ .
$\log(12 \times 10^5) = \log 12 + \log 10^5$
Calculate $\log 12$ using the given values.
Rewrite $12 = \frac{6 \times 10}{5}$ .
$\log 12 = \log\left(\frac{6 \times 10}{5}\right) = \log 6 + \log 10 - \log 5$ $\log 12 = 0.7781 + 1 - 0.6990 = 1.7781 - 0.6990 = 1.0791$
Calculate $\log 10^5$ .
$\log 10^5 = 5 \log 10 = 5 \times 1 = 5$
Add the results from Step 3 and Step 4.
$\log 1\,200\,000 = 1.0791 + 5 = 6.0791$
The value is $\boxed{\text{6.0791}}$ .

It is given that log x = 6 and log y = -2. Evaluate (a) log (xy) (b) log (1/sqrt(x))

Step-by-step solution

Use the logarithm property $\log(AB) = \log A + \log B$ .

Substitute the given values.

Rewrite the expression using exponent rules.

Use the logarithm property $\log(A^B) = B \log A$ .

Substitute the given value for $\log x$ .

Rewrite 16 as a power of 4.

Use the logarithm property $\log(A^B) = B \log A$ .

Simplify the expression.

Use the logarithm property $\log A - \log B = \log\left(\frac{A}{B}\right)$ .

Simplify the fraction inside the logarithm.

Evaluate $\log 10$ (assuming base 10).

Rewrite 30 as a product of 5 and 6.

Use the logarithm property $\log(AB) = \log A + \log B$ .

Substitute the given values and calculate.

Express $1\,200\,000$ in terms of 10 and a number that can be related to 5 and 6.

Use the logarithm property $\log(AB) = \log A + \log B$ .

Calculate $\log 12$ using the given values.

Calculate $\log 10^5$ .

Add the results from Step 3 and Step 4.

Still stuck on this one?

More Mathematics Questions

It is given that log x = 6 and log y = -2. Evaluate (a) log (xy) (b) log (1/sqrt(x))

Step-by-step solution

Use the logarithm property log⁡(AB)=log⁡A+log⁡B\log(AB) = \log A + \log Blog(AB)=logA+logB.

Substitute the given values.

Rewrite the expression using exponent rules.

Use the logarithm property log⁡(AB)=Blog⁡A\log(A^B) = B \log Alog(AB)=BlogA.

Substitute the given value for log⁡x\log xlogx.

Rewrite 16 as a power of 4.

Use the logarithm property log⁡(AB)=Blog⁡A\log(A^B) = B \log Alog(AB)=BlogA.

Simplify the expression.

Use the logarithm property log⁡A−log⁡B=log⁡(AB)\log A - \log B = \log\left(\frac{A}{B}\right)logA−logB=log(BA​).

Simplify the fraction inside the logarithm.

Evaluate log⁡10\log 10log10 (assuming base 10).

Rewrite 30 as a product of 5 and 6.

Use the logarithm property log⁡(AB)=log⁡A+log⁡B\log(AB) = \log A + \log Blog(AB)=logA+logB.

Substitute the given values and calculate.

Express 1 200 0001\,200\,0001200000 in terms of 10 and a number that can be related to 5 and 6.

Use the logarithm property log⁡(AB)=log⁡A+log⁡B\log(AB) = \log A + \log Blog(AB)=logA+logB.

Calculate log⁡12\log 12log12 using the given values.

Calculate log⁡105\log 10^5log105.

Add the results from Step 3 and Step 4.

Still stuck on this one?

More Mathematics Questions

Use the logarithm property $\log(AB) = \log A + \log B$ .

Use the logarithm property $\log(A^B) = B \log A$ .

Substitute the given value for $\log x$ .

Use the logarithm property $\log(A^B) = B \log A$ .

Use the logarithm property $\log A - \log B = \log\left(\frac{A}{B}\right)$ .

Evaluate $\log 10$ (assuming base 10).

Use the logarithm property $\log(AB) = \log A + \log B$ .

Express $1\,200\,000$ in terms of 10 and a number that can be related to 5 and 6.

Use the logarithm property $\log(AB) = \log A + \log B$ .

Calculate $\log 12$ using the given values.

Calculate $\log 10^5$ .