Question

e^{8x} - 20 = 127
enter the correct values in the boxes.
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e^{8x} - 20 = 127
e^{8x} = \square
8x = \ln(\square)
x \approx \square

Explanation:

Step1: Isolate the exponential term

To solve for \( e^{8x} \), we add 20 to both sides of the equation \( e^{8x}-20 = 127 \).

$$ e^{8x}-20 + 20=127 + 20 $$

$$ e^{8x}=147 $$

Step2: Take the natural logarithm

Since \( y = e^{x} \) and \( y=\ln(x) \) are inverse functions, we take the natural logarithm of both sides to solve for \( 8x \).

$$ \ln(e^{8x})=\ln(147) $$

Using the property \( \ln(e^{a})=a \), we get:

$$ 8x = \ln(147) $$

Step3: Solve for \( x \)

We divide both sides by 8 to solve for \( x \).

$$ x=\frac{\ln(147)}{8} $$

Calculating the numerical value: \( \ln(147)\approx 4.9904 \), so \( x\approx\frac{4.9904}{8}\approx 0.6238 \) (or more accurately, using a calculator for \( \frac{\ln(147)}{8} \))

Answer:

For \( e^{8x}=\boxed{147} \)

For \( 8x = \ln(\boxed{147}) \)

For \( x\approx\boxed{0.62} \) (or more precise \( \frac{\ln(147)}{8}\approx0.6238 \))