How do you solve 6^(x-2)=4^x?

Nov 8, 2016

$x = 8.838$

Explanation:

In the equation ${6}^{x - 2} = {4}^{x}$, taking log on both sides we get

$\left(x - 2\right) \log 6 = x \log 4$

or $x \log 6 - 2 \log 6 = x \log 4$

or $x \log 6 - x \log 4 = 2 \log 6$

or $x = \frac{2 \log 6}{\log 6 - \log 4}$

= $\frac{2 \times 0.7782}{0.7782 - 0.6021}$

= $\frac{1.5564}{0.1761} = 8.838$

Nov 8, 2016

$x = 8.838$

Explanation:

${6}^{x - 2} = {4}^{x}$

Log both sides.

$\log {6}^{x - 2} = \log {4}^{x}$

Recall the log rule $\log {x}^{a} = a \log x$

$\left(x - 2\right) \log 6 = x \log 4$

Distribute the $\log 6$

$x \log 6 - 2 \log 6 = x \log 4$

Subtract $x \log 6$ from both sides.

$- 2 \log 6 = x \log 4 - x \log 6$

Factor out the $x$ on the right side

-2log6=x(log4-log6)

Recall the log rule $\log a - \log b = \log \left(\frac{a}{b}\right)$

$- 2 \log 6 = x \log \left(\frac{4}{6}\right)$

$- 2 \log 6 = x \log \left(\frac{2}{3}\right)$

Divide both sides by $\log \left(\frac{2}{3}\right)$

$\frac{- 2 \log 6}{\log \left(\frac{2}{3}\right)} = x$

Use a calculator....

$x = 8.838$