Solve equations of the form:

• \(a^x=b \)


• \(e^{ax+b} = p \)


• \(\ln{(ax+b)} = q \)

Exponential equations

\(a^x=b \quad \) (\(\log\) both sides)

\(\log{a^x}=\log{b} \quad \) (apply the power law)

\(x\log{a}=\log{b} \quad \) (divide by \(\log{a}\))

\(x=\dfrac{\log{b}}{\log{a}} \quad \)

Exponential equations with the natural base

\(e^{ax+b} = p \quad \) (\(\ln\) both sides)

\(\ln{e^{ax+b}}=\ln{p} \quad \) (cancel)

\(ax+b=\ln{p} \quad \) (rearrange)

\(x=\dfrac{\ln{p}-b}{a} \quad \) (rearrange)

Natural logarithmic equations

\(\ln{(ax+b)} = q \quad \) (\(e\) to the power of both sides)

\(e^{\ln{(ax+b)}}=e^{q} \quad \) (cancel)

\(ax+b=e^{q} \quad \) (rearrange)

\(x=\dfrac{e^{q}-b}{a} \quad \) (rearrange)