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Consider the equation [tex]\( 14 \cdot 10^{0.5w} = 100 \)[/tex].

1. Solve the equation for [tex]\( w \)[/tex]. Express the solution as a logarithm in base-10.
[tex]\[
w = \square
\][/tex]

2. Approximate the value of [tex]\( w \)[/tex]. Round your answer to the nearest thousandth.
[tex]\[
w \approx \square
\][/tex]



Answer :

GinnyAnswer
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]

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