Questionb. $e^{x-7}=10000$

Studdy Solution

STEP 1

1. The equation $e^{x-7} = 10000$ is an exponential equation.
2. We will need to isolate the variable $x$ by using the natural logarithm, which is the inverse of the exponential function.

STEP 2

1. Isolate the exponential term $e^{x-7}$ .
2. Use the natural logarithm to solve for $x$ .

STEP 3

The exponential term $e^{x-7}$ is already isolated on one side of the equation:
$e^{x-7} = 10000$

STEP 4

Apply the natural logarithm to both sides of the equation to get the variable out of the exponent. The natural logarithm is the inverse of the exponential function:
$\ln(e^{x-7}) = \ln(10000)$
Using the property of logarithms that $\ln(e^y) = y$ , we simplify the left side:
$x - 7 = \ln(10000)$

STEP 5

Solve for $x$ by adding 7 to both sides:
$x - 7 = \ln(10000)$ $x = \ln(10000) + 7$
The value of $x$ is:
$x = \ln(10000) + 7$
To find the numerical value, calculate $\ln(10000)$ :
$\ln(10000) = \ln(10^4) = 4 \ln(10) \approx 4 \times 2.302 = 9.208$
Thus,
$x \approx 9.208 + 7 = 16.208$
The approximate value of $x$ is:
$\boxed{16.208}$

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