Use the Definition of a Logarithm to Solve Logarithmic Equations
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$$2 + 2 \ln x = 7$$
A) $\left\{ \mathrm { e } ^ { 5 / 2 } \right\}$
B) $\left\{ \frac { e ^ { 5 } } { 2 } \right\}$
C) $\left\{ \ln \frac { 5 } { 2 } \right\}$
D) $\left\{ \frac { 5 } { 2 \ln 1 } \right\}$

Question

Quizplus · Accepted Answer

Using the logarithmic identity $\ln(a^b) = b\ln(a)$, we can rewrite the equation as:
$$2 + \ln(x^2) = 7$$
Subtracting 2 from both sides and simplifying, we get:
$$\ln(x^2) = 5$$
Using the definition of a logarithm, we know that $y = \log_ax$ if and only if $x = a^y$. Applying this to our equation, we get:
$$x^2 = e^5$$
Taking the square root of both sides, we get:
$$x = \pm e^{5/2}$$
However, we need to reject the negative root since the original logarithmic expression is only defined for positive numbers. Therefore, the solution is:
$$x = e^{5/2}$$
which is equivalent to $\left\{\mathrm{e}^{5/2}\right\}$ in set notation.

Use the Definition of a Logarithm to Solve Logarithmic Equations 2+2ln⁡x=72 + 2 \ln x = 72+2lnx=7

Use the Definition of a Logarithm to Solve Logarithmic Equations $2 + 2 \ln x = 7$