Consider the equation 0.75*10^((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~~◻

Isolate 10^(w/3): Isolate 10^(w/3) by dividing both sides by 0.75: 0.75*10^(w/3) = 30 → 10^(w/3) = 30 / 0.75 = 40. Apply logarithm base-10: Apply logarithm base-10 to both sides to solve for w/3: log_10(10^(w/3)) = log_10(40). Simplify using logarithm property: Simplify using the property of logarithms log_b(b^x) = x: w/3 = log_10(40). Solve for w: Solve for w by multiplying both sides by 3: w = 3 * log_10(40). Approximate log_10(40): Use a calculator to approximate log_10(40): log_10(40) ≈ 1.602. Calculate w: Calculate w: w ≈ 3 * 1.602 = 4.806.

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Precalculus

Find the roots of factored polynomials

Full solution

Q. Consider the equation

0.75\times10^{\left(\frac{w}{3}\right)}=30

\newline

Solve the equation for

w

. Express the solution as a logarithm in base

-10

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w=

\square

\newline

Approximate the value of

w

. Round your answer to the nearest thousandth.

\newline

w\approx

\square

Isolate $10^{w/3}$ : Isolate $10^{w/3}$ by dividing both sides by $0.75$ : $0.75\cdot10^{w/3} = 30 \rightarrow 10^{w/3} = \frac{30}{0.75} = 40.$
Apply logarithm base $-10$ : Apply logarithm base $-10$ to both sides to solve for $w/3$ : $\log_{10}(10^{w/3}) = \log_{10}(40)$ .
Simplify using logarithm property: Simplify using the property of logarithms $\log_b(b^x) = x$ : $\frac{w}{3} = \log_{10}(40)$ .
Solve for w: Solve for w by multiplying both sides by $3$ : $w = 3 \times \log_{10}(40)$ .
Approximate $\log_{10}(40)$ : Use a calculator to approximate $\log_{10}(40)$ : $\log_{10}(40) \approx 1.602$ .
Calculate w: Calculate $w$ : $w \approx 3 \times 1.602 = 4.806$ .