Question Find the value of $x$ when $y = a + b \ln x$ , given $a = 12.3301088919$ and $b = -8.22097130319$ , and $y = -16$ .

Studdy Solution

STEP 1

Assumptions
1. The function is given by $y = a + b \ln x$ .
2. The values of $a$ and $b$ are given as $a = 12.3301088919$ and $b = -8.22097130319$ , respectively.
3. We need to find the value of $x$ when $y = -16$ .

STEP 2

We will start by substituting the given values of $a$ , $b$ , and $y$ into the function to create an equation that we can solve for $x$ .
$y = a + b \ln x$

STEP 3

Substitute $y = -16$ , $a = 12.3301088919$ , and $b = -8.22097130319$ into the equation.
$-16 = 12.3301088919 - 8.22097130319 \ln x$

STEP 4

Isolate the term containing $\ln x$ by subtracting $a$ from both sides of the equation.
$-16 - 12.3301088919 = -8.22097130319 \ln x$

STEP 5

Calculate the left-hand side of the equation to simplify it.
$-28.3301088919 = -8.22097130319 \ln x$

STEP 6

Divide both sides of the equation by $b$ to isolate $\ln x$ .
$\frac{-28.3301088919}{-8.22097130319} = \ln x$

STEP 7

Calculate the right-hand side of the equation to find the value of $\ln x$ .
$\ln x = \frac{-28.3301088919}{-8.22097130319}$

STEP 8

Compute the value of $\ln x$ .
$\ln x \approx 3.445$

STEP 9

To find the value of $x$ , we need to exponentiate both sides of the equation, since $\ln x$ is the natural logarithm of $x$ .
$e^{\ln x} = e^{3.445}$

STEP 10

Simplify the left-hand side of the equation using the property $e^{\ln x} = x$ .
$x = e^{3.445}$

STEP 11

Calculate the value of $x$ using a calculator or computational software.
$x \approx 31.37$
The value of $x$ when $y = -16$ is approximately $31.37$ .

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