Question

Find the real solution to the logarithmic equation
$\ln \left(x\right)+\ln \left(2\right)=3$

• A. no real solutions
• B. $0.5\times e^3$
• C. $e^{3/\ln \left(2\right)}$
• D. $3^{e/2}$
• E. $\frac{3}{\ln \left(2\right)}$

Answer 1

The correct option is B $0.5\times e^3$

Let both sides be exponents of the base $e$.

The equation $\ln \left(x\right)+\ln \left(2\right)=3$

$\Rightarrow e^{\ln \left(x\right)+\ln \left(2\right)}=e^3$

$\Rightarrow e^{\ln \left(2x\right)}=e^3$

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 $e^{\ln \left(2x\right)}=e^3$

$\Rightarrow 2x=e^3$

$⟹x=0.5\times e^3$

Hence, option B is correct.