Sure! Let's solve the equation [tex]\( 14 \cdot 10^{0.5 w} = 100 \)[/tex].
### Step-by-Step Solution:
1. Take the logarithm of both sides:
[tex]\[
\log_{10}(14 \cdot 10^{0.5 w}) = \log_{10}(100)
\][/tex]
2. Apply the property of logarithms:
[tex]\[
\log_{10}(14) + \log_{10}(10^{0.5 w}) = \log_{10}(100)
\][/tex]
3. Use the power rule of logarithms, [tex]\(\log_{10}(a^b) = b \cdot \log_{10}(a)\)[/tex]:
[tex]\[
\log_{10}(14) + 0.5 w \cdot \log_{10}(10) = \log_{10}(100)
\][/tex]
4. Note that [tex]\(\log_{10}(10)\)[/tex] is 1:
[tex]\[
\log_{10}(14) + 0.5 w = \log_{10}(100)
\][/tex]
5. Calculate the numerical values:
- [tex]\(\log_{10}(14)\)[/tex] approximately equals 1.146
- [tex]\(\log_{10}(100)\)[/tex] equals 2
6. Substitute these values back into the equation:
[tex]\[
1.146 + 0.5 w = 2
\][/tex]
7. Solve for [tex]\(w\)[/tex]:
[tex]\[
0.5 w = 2 - 1.146
\][/tex]
[tex]\[
0.5 w = 0.854
\][/tex]
[tex]\[
w = \frac{0.854}{0.5}
\][/tex]
[tex]\[
w = 1.708
\][/tex]
### Final Answer:
#### Exact form:
[tex]\[
w = 2 \cdot (\log_{10}(100) - \log_{10}(14))
\][/tex]
#### Approximate form:
[tex]\[
w \approx 1.708
\][/tex]
