Question

Consider the equation 14 x 10^0.5w= 100.Solve the equation for w. Express the solution as a logarithm in base-10.W = _____ Approximate the value of w. Round your answer to the nearest thousandth.w ≈ _____

Answer 1

Solution:

Given;

$14\cdot10^{0.5w}=100$

Divide both sides by 14, we have;

$\begin{gathered} \frac{14\cdot10^{0.5w}}{14}=\frac{100}{14} \\ \\ 10^{0.5w}=\frac{50}{7} \end{gathered}$

Take the logarithm of both sides; we have;

$\log_{10}(10)^{0.5w}=\log_{10}(\frac{50}{7})$

Applying logarithmic laws;

$0.5w=\log_{10}(\frac{50}{7})$

Divide both sides by 0.5;

$\begin{gathered} \frac{0.5w}{0.5}=\frac{\log_{10}(\frac{50}{7})}{0.5} \\ \\ w=\frac{\operatorname{\log}_{10}(\frac{50}{7})}{0.5} \end{gathered}$

(b)

$\begin{gathered} w=\frac{\operatorname{\log}_{10}(\frac{50}{7})}{0.5} \\ \\ w\approx1.708 \end{gathered}$