How do you simplify ln((5e^x)-(10e^2x))?

Jan 2, 2016

If you meant $\ln \left(\left(5 {e}^{x}\right) - \left(10 {e}^{2 x}\right)\right)$

Then you can factor the ${e}^{x}$ and use $\ln \left(a \cdot b\right) = \ln a + \ln b$

$x + \ln 5 + \ln \left(1 - 2 {e}^{x}\right)$

Explanation:

It can't actually. You can't simplify polynomials with exponential functions. The fact that it is substraction (and not multiplication or division) leaves no room for simplifications.

However, if you meant $\ln \left(\left(5 {e}^{x}\right) - \left(10 {e}^{2 x}\right)\right)$

$\ln \left(5 {e}^{x} - 10 {e}^{x} \cdot {e}^{x}\right)$

Factor the $5 {e}^{x}$:

$\ln \left(5 \cdot {e}^{x} \cdot \left(1 - 2 {e}^{x}\right)\right)$

Use of the property $\ln \left(a \cdot b \cdot c\right) = \ln a + \ln b + \ln c$ gives:

$\ln 5 + \ln {e}^{x} + \ln \left(1 - 2 {e}^{x}\right)$

Since $\ln = {\log}_{e}$

$\ln 5 + x + \ln \left(1 - 2 {e}^{x}\right)$