2*5^x = 3^(2x) — step-by-step solution

1) Let's solve in

ℝ

2 \cdot 5^{x} = 3^{2 x}

A power of a power multiplies the exponents: to have the same exponent on both sides, we write

3^{2 x}

{(3^{2})}^{x}

9^{x}

2 \cdot 5^{x} = 9^{x}

ⓘ We divide both sides by

5^{x}

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

\frac{2 \cdot 5^{x}}{5^{x}} = \frac{9^{x}}{5^{x}}

, that is

2 = {(\frac{9}{5})}^{x}

{(\frac{9}{5})}^{x} = 2

Let's apply the ln function on both sides:

\ln ({(\frac{9}{5})}^{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}{5})}^{x})

x \cdot \ln (\frac{9}{5})

x \cdot \ln (\frac{9}{5}) = \ln (2)

ⓘ Both sides are divided by

\ln (\frac{9}{5})

▶Solution obtained:

x = \frac{\ln (2)}{\ln (\frac{9}{5})}

2) The numeric value of:

x = \frac{\ln (2)}{\ln (\frac{9}{5})}

is:

x = \frac{\approx 0.6931}{\ln (1.8)}

x = \frac{\approx 0.6931}{\approx 0.5878}

▷

x = \approx 1.1792