QuestionCheck for extraneous solutions.
8. $\ln 2 a=9$
2

Studdy Solution

STEP 1

1. The equation $\ln 2a = 9$ involves the natural logarithm.
2. We need to solve for $a$ and check if the solution is valid in the context of the original equation.

STEP 2

1. Solve the logarithmic equation for $a$ .
2. Check for extraneous solutions.

STEP 3

To solve for $a$ , we first need to rewrite the logarithmic equation in its exponential form. Recall that if $\ln b = c$ , then $b = e^c$ . Apply this to the equation:
$\ln 2a = 9$
Rewrite it as:
$2a = e^9$

STEP 4

Next, solve for $a$ by dividing both sides by 2:
$2a = e^9$ $a = \frac{e^9}{2}$

STEP 5

Check for extraneous solutions. Since the natural logarithm function is defined for positive arguments, we need to ensure that $2a > 0$ . Given $a = \frac{e^9}{2}$ , we have:
$2a = e^9$
Since $e^9$ is positive, $2a$ is also positive, confirming that the solution is valid and not extraneous.
The solution for $a$ is:
$\boxed{\frac{e^9}{2}}$

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