How do you solve e^(p+10)+4=18?

Aug 17, 2016

$p = \ln \left(14\right) - 10 \approx - 7.3609$

Explanation:

The base-$a$ logarithm of $x$, denoted ${\log}_{a} \left(x\right)$, is the power which $a$ must be taken to for ${a}^{{\log}_{a} \left(x\right)}$ to equal $x$. Because of this, it should be quite clear that for any $x$, we have ${\log}_{a} \left({a}^{x}\right) = x$, as $x$ is the power which $a$ must be taken to to obtain ${a}^{x}$.

Using this, along with the convention that the natural log $\ln \left(x\right)$ is the base-$e$ logarithm of $x$:

${e}^{p + 10} + 4 = 18$

$\implies {e}^{p + 10} = 14$

$\implies \ln \left({e}^{p + 10}\right) = \ln \left(14\right)$

$\implies p + 10 = \ln \left(14\right)$

$\therefore p = \ln \left(14\right) - 10 \approx - 7.3609$