Question

consider the equation $0.75 \cdot 10^{\frac{w}{3}} = 30$. 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 \approx $

Explanation:

Step1: Isolate the exponential term

Divide both sides by 0.75:
$$10^{\frac{w}{3}} = \frac{30}{0.75}$$
$$10^{\frac{w}{3}} = 40$$

Step2: Convert to base-10 logarithm

Apply $\log_{10}$ to both sides:
$$\frac{w}{3} = \log_{10}(40)$$

Step3: Solve for $w$

Multiply both sides by 3:
$$w = 3\log_{10}(40)$$

Step4: Approximate the value

Calculate $\log_{10}(40) \approx 1.60206$, then multiply by 3:
$$w \approx 3 \times 1.60206 = 4.80618$$
Round to nearest thousandth.

Answer:

$w = 3\log_{10}(40)$
$w \approx 4.806$