State the domain, then combine logarithms of the same base. log_a M=log_a N M=N log_a M=k M=ak Here a>0, a 1 and M,N>0. Reject any root that makes an original argument non-positive or a base invalid. Exponential equations ax=b x= ln bln a (a>0, a 1, b>0) 3x=20 x= [[fill]] ln20ln3 2.73[[/fill]] (3 s.f.) For equations in a2x and ax, let u=ax>0. Solve the quadratic in u, then find x.

State the domain, then combine logarithms of the same base.

Compare arguments

Convert to index form

Reject any root that makes an original argument non-positive or a base invalid.

Check the original equation

\(a^x=b\quad\Longrightarrow\quad x=\dfrac{\ln b}{\ln a}\quad(a>0,\ a\ne1,\ b>0)\)

\(3^x=20\quad\Longrightarrow\quad x=\) [[fill]]\(\dfrac{\ln20}{\ln3}\approx2.73\)[[/fill]] \(\quad(3\text{ s.f.})\)

For equations in \(a^{2x}\) and \(a^x\), let \(u=a^x>0\). Solve the quadratic in \(u\), then find \(x\).

Solving Logarithmic and Exponential Equations

Solving Logarithmic and Exponential Equations