Full solution

Q. Consider the equation

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

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Solve the equation for

x

. Express the solution as a logarithm in base

10

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x=

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Approximate the value of

x

. Round your answer to the nearest thousandth.

\newline

x \approx

Isolate the exponential term: Isolate the exponential term. $\newline$ To solve for $x$ , we first want to isolate the term with the exponent. We can do this by dividing both sides of the equation by $4$ . $\newline$ $4 \cdot 10^{(-3x)} = 18$ $\newline$ $10^{(-3x)} = \frac{18}{4}$ $\newline$ $10^{(-3x)} = 4.5$
Apply the logarithm: Apply the logarithm to both sides of the equation. $\newline$ To solve for the exponent, we can take the logarithm of both sides of the equation. We will use the common logarithm (base $10$ ). $\newline$ $\log(10^{(-3x)}) = \log(4.5)$
Use logarithm property: Use the property of logarithms to bring down the exponent. $\newline$ The property of logarithms that we will use is $\log(b^a) = a \cdot \log(b)$ . This allows us to move the exponent in front of the logarithm. $\newline$ $-3x \cdot \log(10) = \log(4.5)$
Simplify the equation: Simplify the left side of the equation. $\newline$ Since $\log(10)$ is equal to $1$ , the equation simplifies to: $\newline$ $-3x = \log(4.5)$
Solve for x: Solve for x. $\newline$ To solve for x, we divide both sides of the equation by $-3$ . $\newline$ $x = \frac{\log(4.5)}{-3}$
Approximate the value of x: Approximate the value of x. $\newline$ Using a calculator, we can find the value of $\log(4.5)$ and then divide by $-3$ to get the approximate value of x. $\newline$ $x \approx \frac{\log(4.5)}{-3}$ $\newline$ $x \approx \frac{0.6532}{-3}$ $\newline$ $x \approx -0.2177$ $\newline$ Rounded to the nearest thousandth, $x \approx -0.218$