# How do you solve ln x + ln 5 = ln10?

Nov 29, 2015

$x = 2$

#### Explanation:

Subtract $\ln \left(5\right)$ from both sides:

$\ln \left(x\right) = \ln \left(10\right) - \ln \left(5\right)$

Use the rule $\ln \left(a\right) - \ln \left(b\right) = \ln \left(\frac{a}{b}\right)$:

$\ln \left(x\right) = \ln \left(\frac{10}{5}\right) = \ln \left(2\right)$.

Since the logarithm is injective, $x$ must be $2$. If you want to see it in an other way, take the exponential on both sides:

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

Since logarithm and exponential are inverse functions, we have that ${e}^{\ln \left(z\right)} = z$. So,

$x = 2$