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Consider the equation

-16*10^(6x)=-80". "
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~~

Consider the equation $\newline$ $-16 \times 10^{6x} = -80$ . $\newline$ Solve the equation for $\newline$ $x$ . Express the solution as a logarithm in base $-10$ . $\newline$ $x=$ $\newline$ Approximate the value of $\newline$ $x$ . Round your answer to the nearest thousandth. $\newline$ $x \approx$

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Properties of logarithms: mixed review

Full solution

Q. Consider the equation

\newline

-16 \times 10^{6x} = -80

.

\newline

Solve the equation for

\newline

x

. Express the solution as a logarithm in base

-10

.

\newline

x=

\newline

Approximate the value of

\newline

x

. Round your answer to the nearest thousandth.

\newline

x \approx

Identify Equation: Identify the equation that needs to be solved. $\newline$ The equation given is $-16\times10^{6x} = -80$ .
Isolate Exponential Term: Isolate the exponential term. $\newline$ To isolate the exponential term, divide both sides of the equation by \(-16＂). $\newline$ \(-16\cdot $10$ ^{( $6$ x)} / $-16$ = $-80$ / $-16$ ＂) $\newline$ \(10^{( $6$ x)} = $5$ ＂)
Apply Logarithm: Apply the logarithm to both sides of the equation. $\newline$ To solve for $x$ , take the logarithm of both sides of the equation. We will use the base- $10$ logarithm since the exponential base is $10$ . $\newline$ $\log(10^{6x}) = \log(5)$
Use Power Property: Use the power property of logarithms. $\newline$ The power property of logarithms states that $\log_b(m^n) = n \cdot \log_b(m)$ . Apply this property to the left side of the equation. $\newline$ $6x \cdot \log(10) = \log(5)$
Simplify Equation: Simplify the left side of the equation. $\newline$ Since $\log(10)$ is equal to $1$ , the equation simplifies to: $\newline$ $6x = \log(5)$
Solve for x: Solve for x. $\newline$ Divide both sides of the equation by $6$ to solve for $x$ . $\newline$ $x = \frac{\log(5)}{6}$
Approximate Value: Approximate the value of $x$ . Use a calculator to find the value of $\log(5)$ and then divide by $6$ . $x \approx \log(5) / 6$ $x \approx 0.69897 / 6$ $x \approx 0.116495$ Round the answer to the nearest thousandth. $x \approx 0.116$

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