Certainly! Let's work through the steps to find [tex]\(\log_{10} 36^{1/3}\)[/tex] given the approximations [tex]\(\log_{10} 2 = 0.3010\)[/tex] and [tex]\(\log_{10} 3 = 0.4771\)[/tex].
We need to compute [tex]\(\log_{10} 36^{1/3}\)[/tex]. Let's break it down step by step using properties of logarithms.
### Step 1: Transform the original expression using logarithm properties.
We know:
[tex]\[
\log_{10} 36^{1/3} = \frac{1}{3} \log_{10} 36
\][/tex]
So we need to find [tex]\(\log_{10} 36\)[/tex] first.
### Step 2: Express 36 in terms of its factors.
The number 36 can be written as:
[tex]\[
36 = 6^2
\][/tex]
Using the logarithm property [tex]\(\log_{10} (a^b) = b \log_{10} a\)[/tex], we get:
[tex]\[
\log_{10} 36 = \log_{10} 6^2 = 2 \log_{10} 6
\][/tex]
### Step 3: Express 6 in terms of its prime factors.
The number 6 can be expressed as:
[tex]\[
6 = 2 \times 3
\][/tex]
Using the logarithm property [tex]\(\log_{10} (ab) = \log_{10} a + \log_{10} b\)[/tex], we get:
[tex]\[
\log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3
\][/tex]
### Step 4: Substitute the given values into the expression.
We know:
[tex]\[
\log_{10} 2 = 0.3010 \quad \text{and} \quad \log_{10} 3 = 0.4771
\][/tex]
So,
[tex]\[
\log_{10} 6 = 0.3010 + 0.4771 = 0.7781
\][/tex]
### Step 5: Substitute back to find [tex]\(\log_{10} 36\)[/tex].
From Step 2, we have:
[tex]\[
\log_{10} 36 = 2 \log_{10} 6 = 2 \times 0.7781 = 1.5562
\][/tex]
### Step 6: Finally, find [tex]\(\log_{10} 36^{1/3}\)[/tex].
From Step 1, we need:
[tex]\[
\log_{10} 36^{1/3} = \frac{1}{3} \log_{10} 36 = \frac{1}{3} \times 1.5562 = 0.5187\overline{3}
\][/tex]
Thus, the logarithm [tex]\(\log_{10} 36^{1/3}\)[/tex] is approximately [tex]\(0.5187\overline{3}\)[/tex].
