Question

Find all the real solutions to the logarithmic equation
$\ln \left(x^4\right)+\ln \left(x^2\right)-\ln \left(x^3\right)-2=7$

• A. $9^{1/3}$
• B. $e^3$
• C. $1$
• D. no real solutions
• E. $0$

Answer 1

The correct option is B $e^3$

Let both sides be exponents of the base e. The equation $\ln \left(x^4\right)+\ln \left(x^2\right)-\ln \left(x^3\right)-2=7$ that is $\ln \left(x^4\right)+\ln \left(x^2\right)-\ln \left(x^3\right)=9$can be rewritten as $e^{\ln \left(x^4\right)+\ln \left(x^2\right)-\ln \left(x^3\right)}=e^9$ or $e^{\ln (x^4\ast x^2/(x^3\left)\right)}=e^9$ that is $e^{\ln \left(x^3\right)}=e^9$.

By now you should know that when the base of the exponent and the base of the logarithm are the same, the left side can be written $x$. The equation $e^{\ln \left(x^3\right)}=e^2$ can now be written as $x^3=e^9$ that is

$x={\left(e^9\right)}^{1/3}$

$x=e^3$

Therefore, $x=e^3$.

Hence option B is correct.