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100*2^(4x)=15
What is the solution of the equation?
Round your answer, if necessary, to the nearest thousandth.

x~~

$100 \cdot 2^{4 x}=15$ $\newline$ What is the solution of the equation? $\newline$ Round your answer, if necessary, to the nearest thousandth. $\newline$ $x \approx$

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Solve exponential equations using logarithms

Full solution

Q.

100 \cdot 2^{4 x}=15

\newline

What is the solution of the equation?

\newline

Round your answer, if necessary, to the nearest thousandth.

\newline

x \approx

Isolate exponential term: First, we need to isolate the exponential term by dividing both sides of the equation by $100$ . $\newline$ $100 \cdot 2^{4x} = 15$ $\newline$ $2^{4x} = \frac{15}{100}$ $\newline$ $2^{4x} = 0.15$
Apply logarithm: Now, we apply the logarithm to both sides of the equation to solve for $x$ . We can use the natural logarithm ( $\ln$ ) or the common logarithm ( $\log$ ). Here, we'll use the natural logarithm for convenience. $\newline$ $\ln(2^{4x}) = \ln(0.15)$
Move exponent in front: Using the power property of logarithms, we can move the exponent in front of the logarithm. $4x \cdot \ln(2) = \ln(0.15)$
Solve for x: Next, we solve for $x$ by dividing both sides of the equation by $4\cdot\ln(2)$ .
$x = \frac{\ln(0.15)}{4 \cdot \ln(2)}$
Calculate x: Now, we calculate the value of x using a calculator. $\newline$ $x \approx \frac{\ln(0.15)}{4 \cdot \ln(2)}$ $\newline$ $x \approx \frac{-1.897119984}{4 \cdot 0.693147181}$ $\newline$ $x \approx \frac{-1.897119984}{2.772588724}$ $\newline$ $x \approx -0.684$

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