To evaluate the expression [tex]\(\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}\)[/tex] using logarithms, we can break it down into a series of steps. Here is a detailed, step-by-step solution:
1. Identify the components of the expression:
The given expression is:
[tex]\[
\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}
\][/tex]
2. Use logarithms to simplify the expression:
We know that using logarithm properties can help simplify the calculations. Recall the properties:
[tex]\[
\log(a^b) = b \log(a)
\][/tex]
and
[tex]\[
\log\left(\frac{a \cdot b}{c}\right) = \log(a) + \log(b) - \log(c)
\][/tex]
3. Break down the expression using logarithms:
Let:
[tex]\[
a = 17.2, \quad b = 4.93, \quad c = 6750000, \quad \text{and} \quad d = \frac{1}{3}
\][/tex]
We need to find:
[tex]\[
\log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right)
\][/tex]
Using the properties of logarithms:
[tex]\[
\log\left(17.2^2 \times 4.93\right) = \log(17.2^2) + \log(4.93)
\][/tex]
So,
[tex]\[
\log(17.2^2) = 2 \log(17.2)
\][/tex]
Now we rewrite the logarithm of the denominator using the property of exponents:
[tex]\[
\log(6750000^{1/3}) = \frac{1}{3} \log(6750000)
\][/tex]
4. Calculate individual logarithms using values:
From the logarithm tables or given data:
[tex]\[
\log(17.2) = 1.235528446907549
\][/tex]
[tex]\[
\log(4.93) = 0.69284691927723
\][/tex]
[tex]\[
\log(6750000) = 6.829303772831025
\][/tex]
Therefore:
[tex]\[
2 \log(17.2) = 2 \times 1.235528446907549 = 2.471056893815098
\][/tex]
[tex]\[
\frac{1}{3} \log(6750000) = \frac{1}{3} \times 6.829303772831025 = 2.276434590943675
\][/tex]
5. Combine the logarithmic terms:
Now, put these values into the expression:
[tex]\[
\log\left(\frac{17.2^2 \times 4.93}{6750000^{1/3}}\right) = 2.471056893815098 + 0.69284691927723 - 2.276434590943675
\][/tex]
Simplify the right side:
[tex]\[
2.471056893815098 + 0.69284691927723 - 2.276434590943675 = 0.8874692221486531
\][/tex]
6. Convert back from logarithms to find the result:
To find the final result, convert [tex]\(\log(x) = 0.8874692221486531\)[/tex] back to [tex]\(x\)[/tex]:
[tex]\[
x = 10^{0.8874692221486531}
\][/tex]
Therefore:
[tex]\[
x \approx 7.7173682172212095
\][/tex]
Hence, the value of [tex]\(\frac{17.2^2 \times 4.93}{\sqrt[3]{6750000}}\)[/tex] evaluates to approximately [tex]\(7.7173682172212095\)[/tex].
