# Question #3624b

Jun 13, 2017

$x = {\log}_{\frac{25}{32}} 500$

#### Explanation:

${5}^{2 x - 3} - {2}^{5 x + 2} = 0$
${5}^{2 x - 3} = {2}^{5 x + 2}$
Taking the natural logarithm of both sides,
$\ln \left({5}^{2 x - 3}\right) = \ln \left({2}^{5 x + 2}\right)$
$\left(2 x - 3\right) \ln 5 = \left(5 x + 2\right) \ln 2$
$\left(2 \ln 5\right) x - 3 \ln 5 = \left(5 \ln 2\right) x + 2 \ln 2$
$\left(2 \ln 5\right) x - \left(5 \ln 2\right) x = 3 \ln 5 + 2 \ln 2$
$\left[\ln \left({5}^{2}\right) - \ln \left({2}^{5}\right)\right] x = \ln \left({5}^{3}\right) + \ln \left({2}^{2}\right)$
$x = \ln \frac{{5}^{3} \cdot {2}^{2}}{\ln} \left({5}^{2} / {2}^{5}\right)$
$x = \ln \frac{125 \cdot 4}{\ln} \left(\frac{25}{32}\right)$
$x = {\log}_{\frac{25}{32}} 500$

Jun 13, 2017

Given: ${5}^{2 x - 3} - {2}^{5 x + 2} = 0$

Move the second term to the right:

${5}^{2 x - 3} = {2}^{5 x + 2}$

Use the base 5 logarithm on both sides:

${\log}_{5} \left({5}^{2 x - 3}\right) = {\log}_{5} \left({2}^{5 x + 2}\right)$

Use the identity ${\log}_{b} \left({a}^{c}\right) = \left(c\right) {\log}_{b} \left(a\right)$ on both sides:

$\left(2 x - 3\right) {\log}_{5} \left(5\right) = \left(5 x + 2\right) {\log}_{5} \left(2\right)$

Use the property ${\log}_{b} \left(b\right) = 1$ on the left:

$2 x - 3 = \left(5 x + 2\right) {\log}_{5} \left(2\right)$

Use the distributive property on the right:

$2 x - 3 = 5 {\log}_{5} \left(2\right) x + 2 {\log}_{5} \left(2\right)$

Subtract $5 {\log}_{5} \left(2\right) x$ from both sides:

$\left(2 - 5 {\log}_{5} \left(2\right)\right) x - 3 = 2 {\log}_{5} \left(2\right)$

$\left(2 - 5 {\log}_{5} \left(2\right)\right) x = 3 + 2 {\log}_{5} \left(2\right)$

Divide both sides by the coefficient of x:

$x = \frac{3 + 2 {\log}_{5} \left(2\right)}{2 - 5 {\log}_{5} \left(2\right)}$

Convert to base e by using the conversion formula ${\log}_{5} \left(x\right) = \ln \frac{x}{\ln} \left(5\right)$:

$x = \frac{3 + 2 \ln \frac{2}{\ln} \left(5\right)}{2 - 5 \ln \frac{2}{\ln} \left(5\right)}$

Multiply by 1 in the form of $\ln \frac{5}{\ln} \left(5\right)$:

$x = \frac{3 \ln \left(5\right) + 2 \ln \left(2\right)}{2 \ln \left(5\right) - 5 \ln \left(2\right)}$