Solve for w 14*10^(0.5w)=100

$14 \cdot 10^{0.5 w} = 100$

Step 1

Divide each term in $(14 \cdot 10^{0.5 w}) = 100$ by $14$ and simplify.

Divide each term in $(14 \cdot 10^{0.5 w}) = 100$ by $14$ .

$\frac{(14 \cdot 10^{0.5 w})}{14} = \frac{100}{14}$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 \cdot 10^{0.5 w}}{14} = \frac{100}{14}$

Divide $10^{0.5 w}$ by $1$ .

$10^{0.5 w} = \frac{100}{14}$

Simplify the right side.

Cancel the common factor of $100$ and $14$ .

Factor $2$ out of $100$ .

$10^{0.5 w} = \frac{2 (50)}{14}$

Cancel the common factors.

Factor $2$ out of $14$ .

$10^{0.5 w} = \frac{2 \cdot 50}{2 \cdot 7}$

Cancel the common factor.

$10^{0.5 w} = \frac{2 \cdot 50}{2 \cdot 7}$

Rewrite the expression.

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

Step 2

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 3

Expand the left side.

Expand $\log (10^{0.5 w})$ by moving $0.5 w$ outside the logarithm.

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

Logarithm base $10$ of $10$ is $1$ .

$0.5 w \cdot 1 = \log (\frac{50}{7})$

Multiply $0.5$ by $1$ .

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

Step 4

Divide each term in $0.5 w = \log (\frac{50}{7})$ by $0.5$ and simplify.

Divide each term in $0.5 w = \log (\frac{50}{7})$ by $0.5$ .

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

Simplify the left side.

Cancel the common factor of $0.5$ .

Cancel the common factor.

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

Divide $w$ by $1$ .

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

Simplify the right side.

Divide $50$ by $7$ .

$w = \frac{\log (7.\overline{142857})}{0.5}$

Log base $10$ of $7.\overline{142857}$ is approximately $0.85387196$ .

$w = \frac{0.85387196}{0.5}$

Divide $0.85387196$ by $0.5$ .

$w = 1.70774392$

Solve for w 14*10^(0.5w)=100

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