inverse of 4^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 = 4^{x}

⇥1) Let's solve in

ℝ

4^{x} = y

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

y > 0

⇥

\ln (4^{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 (4^{x})

x \cdot \ln (4)

⇥

x \cdot \ln (4) = \ln (y)

⇥ⓘ The logarithm of a power is a multiple of the logarithm, ln(a^n) = n*ln(a):

\ln (4)

\ln (2^{2})

2 \cdot \ln (2)

⇥

x \cdot 2 \cdot \ln (2) = \ln (y)

⇥ⓘ Both sides are divided by

2 \cdot \ln (2)

⇥Solution obtained:

x = \frac{\ln (y)}{2 \cdot \ln (2)}

⇥2) Solving for y:

y > 0

⇥Solution found:

y > 0

Renaming the output as

x

, the inverse is

\frac{\ln (x)}{2 \cdot \ln (2)}

, with domain

x > 0

, the condition found when isolating the input

▶Inverse function:

\frac{\ln (x)}{2 \cdot \ln (2)}