Solving Logarithm Equations With a Change of Base

If a log equation involves 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
\[log_3x+log_5x=2\]
.
\[log_5x=\frac{log3}{log5}log_3x\]

\[log_3x+\frac{log3}{log5}log_3x=2\]

\[log_3x(1++\frac{log3}{log5})=2\]

\[log_3x=\frac{2}{(1++\frac{log3}{log5})}=\frac{2log5}{log3+log5}=\frac{log25}{log15}\]

\[x=3^{\frac{log25}{log15}}\]

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