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Consider the equation $4\cdot 10^{-3x}=18$ . Solve the equation for $x$ . Express the solution as a logarithm in base- $10$ . $x=$ Approximate the value of $x$ . Round your answer to the nearest thousandth. $x\approx$

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Properties of logarithms: mixed review

Full solution

Q. Consider the equation

4\cdot 10^{-3x}=18

. Solve the equation for

x

. Express the solution as a logarithm in base-

10

.

x=

Approximate the value of

x

. Round your answer to the nearest thousandth.

x\approx

Divide and Isolate Exponential Term: First, divide both sides by $4$ to isolate the exponential term. $\newline$ $\frac{4 \cdot 10^{-3x}}{4} = \frac{18}{4}$ $\newline$ $10^{-3x} = 4.5$
Take Logarithm of Both Sides: Take the logarithm base $-10$ of both sides to solve for $x$ . $\newline$ $\log_{10}(10^{-3x}) = \log_{10}(4.5)$
Apply Power Rule of Logarithms: Use the power rule of logarithms: $\log_{10}(a^b) = b \cdot \log_{10}(a)$ . $\newline$ $-3x \cdot \log_{10}(10) = \log_{10}(4.5)$
Simplify the Equation: Since $\log_{10}(10) = 1$ , simplify the equation. $\newline$ $-3x = \log_{10}(4.5)$
Solve for x: Solve for $x$ by dividing both sides by $-3$ . $\newline$ $x = \frac{\log_{10}(4.5)}{-3}$
Approximate Logarithm Value: Approximate the value of $\log_{10}(4.5)$ using a calculator. $\newline$ $\log_{10}(4.5) \approx 0.653$
Calculate x: Calculate $x$ by dividing $0$ . $653$ by $-3$ . $\newline$ $x \approx \frac{0.653}{-3} \approx -0.218$

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