If $$ \log 5=0.6990, \log 7=0.8451 $$ and $$ \log 8=0.9031 $$, evaluate $$ \log \left(\frac{35 imes 49}{40 \div 56} ight) $$

Question

If $$ \log 5=0.6990, \log 7=0.8451 $$ and $$ \log 8=0.9031 $$, evaluate $$ \log \left(\frac{35 	imes 49}{40 \div 56}ight) $$

UpStudy AI · Accepted Answer

To evaluate $$ \log \left(\frac{35 \times 49}{40 \div 56}\right) $$, we start by simplifying the expression inside the logarithm.

First, we simplify $$ 40 \div 56 $$:
$$
40 \div 56 = \frac{40}{56} = \frac{5}{7}
$$

Now we can rewrite the original expression:
$$
\frac{35 \times 49}{40 \div 56} = \frac{35 \times 49}{\frac{5}{7}} = 35 \times 49 \times \frac{7}{5}
$$

Next, we can simplify this further:
$$
35 \times 49 \times \frac{7}{5} = \frac{35 \times 49 \times 7}{5}
$$

Now we can calculate $$ 35 \times 49 \times 7 $$:
$$
35 = 5 \times 7
$$
Thus,
$$
35 \times 49 = (5 \times 7) \times 49 = 5 \times 7 \times 49
$$

Now substituting this back into our expression:
$$
\frac{35 \times 49 \times 7}{5} = \frac{(5 \times 7 \times 49) \times 7}{5} = 7 \times 49 \times 7 = 7^2 \times 49
$$

Next, we know that $$ 49 = 7^2 $$, so:
$$
7^2 \times 49 = 7^2 \times 7^2 = 7^4
$$

Now we can express the logarithm:
$$
\log \left(\frac{35 \times 49}{40 \div 56}\right) = \log(7^4)
$$

Using the logarithmic identity $$ \log(a^b) = b \log a $$:
$$
\log(7^4) = 4 \log 7
$$

Now we substitute the value of $$ \log 7 $$:
$$
\log 7 = 0.8451
$$
Thus,
$$
4 \log 7 = 4 \times 0.8451 = 3.3804
$$

Therefore, the final result is:
$$
\log \left(\frac{35 \times 49}{40 \div 56}\right) = 3.3804
$$

The answer is:
$$
\boxed{3.3804}
$$
The simplified answer is 3.3804.

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