9^x = 2e^(x) — step-by-step solution

Let's solve in

ℝ

9^{x} = 2 e^{x}

ⓘ We divide both sides by

e^{x}

, which is strictly positive, the quotient of two powers with the same exponent being the power of the quotient:

\frac{9^{x}}{e^{x}} = \frac{2 e^{x}}{e^{x}}

, that is

{(\frac{9}{e})}^{x} = 2

{(\frac{9}{e})}^{x} = 2

You can apply the ln function on both sides:

\ln ({(\frac{9}{e})}^{x}) = \ln (2)

ⓘ 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 ({(\frac{9}{e})}^{x})

x \cdot \ln (\frac{9}{e})

x \cdot \ln (\frac{9}{e}) = \ln (2)

ⓘ Both sides are divided by

\ln (\frac{9}{e})

▶Solution found:

x = \frac{\ln (2)}{\ln (\frac{9}{e})}

The numeric value of:

x = \frac{\ln (2)}{\ln (\frac{9}{e})}

equals:

x = \frac{\approx 0.6931}{\ln (\frac{9}{\approx 2.7183})}

x = \frac{\approx 0.6931}{\ln (\approx 3.3109)}

x = \frac{\approx 0.6931}{\approx 1.1972}

▷

x = \approx 0.5790