## Log Equations With Change of Base

If terms in a log equation have different bases, then we can't solve the equation without making the bases the same. We can do this using the change of base rule

$log_a x = \frac{log_b}{log_a}log_bx$
.
Example: Solve the equation
$2log_3x+4log_5x=2$
.
$2log_3x+4log_5x=2$
.
$log_5x=\frac{log3}{log5}log_3x$

$2log_3x+4 \frac{log3}{log5}log_3x=2$

$log_3x(2+4\frac{log3}{log5})=2$

$log_3x=\frac{2}{2+4\frac{log3}{log5}}=\frac{2log5}{2log5+4log3}=\frac{log25}{log(5^2 \times 3^4)}$

$x=3^{\frac{log25}{log2025}}$