inverse of 0.3^x — step-by-step solution

An exponential with a positive base other than 1 and a nonconstant affine exponent is strictly monotone on ℝ and takes every strictly positive value

Express input

x

in terms of output

y

y = {0.3}^{x}

⇥Let's solve in

ℝ

{0.3}^{x} = y

⇥The logarithm being defined only for positive numbers, let's apply the ln function on both sides, provided that

y > 0

⇥

\ln ({0.3}^{x}) = \ln (y)

⇥ⓘ On the real domain of the original expression, the base is strictly positive: the logarithm of the power is the exponent times the logarithm of the base:

\ln ({0.3}^{x})

x \cdot \ln (0.3)

⇥

x \cdot \ln (0.3) = \ln (y)

⇥ⓘ Both sides are divided by

\ln (0.3)

⇥Solution found:

x = \frac{\ln (y)}{\ln (0.3)}

⇥Let's solve for y:

y > 0

⇥Solution found:

y > 0

Renaming the output as

x

, the inverse is

\frac{\ln (x)}{\ln (0.3)}

, with domain

x > 0

, the condition found when isolating the input

▶Inverse function:

\frac{\ln (x)}{\ln (0.3)}