# How do you solve 10^(4x-1) = 5000?

Mar 2, 2018

$x = \ln \frac{5000}{1 \ln \left(4\right)} + \frac{1}{4}$

#### Explanation:

Apply the natural logarithm to both sides:

$\ln \left({10}^{4 x - 1}\right) = \ln \left(5000\right)$

$\ln \left({10}^{4 x - 1}\right) = \left(4 x - 1\right) \ln \left(10\right) ,$ as the exponent property for logarithms tells us that $\ln \left({x}^{y}\right) = y \ln \left(x\right)$.

$\ln \left({10}^{4 x - 1}\right) = \ln \left(5000\right) \Leftrightarrow \left(4 x - 1\right) \ln \left(10\right) = \ln \left(5000\right)$

Solve for $x :$

$\frac{\left(4 x - 1\right) \cancel{\ln} \left(10\right)}{\cancel{\ln}} \left(10\right) = \ln \frac{5000}{\ln} \left(10\right)$

$4 x \cancel{- 1 + 1} = \ln \frac{5000}{\ln} \left(10\right) + 1$

$\frac{\cancel{4} x}{\cancel{4}} = \ln \frac{5000}{4 \ln 10} + \frac{1}{4}$