An exponential equation is an equation in which the variable appears in an exponent.
These equations are usually solved by rewriting both sides with the same base or by using logarithms.
Some mixed exponential-linear equations need numerical methods or the Lambert \(W\)-function.
1. What is an exponential equation?
Examples include:
\[
\begin{aligned}
2^x&=16,\\
3^{2x-1}&=27,\\
e^{2x-1}&=5,\\
e^x&=3x.
\end{aligned}
\]
The base of a real exponential expression must usually be positive. In the standard form
\(a^u\), the base must satisfy:
\[
\begin{aligned}
a>0,\qquad a\ne1.
\end{aligned}
\]
2. One-to-one property
For \(a>0\) and \(a\ne1\), the exponential function \(a^x\) is one-to-one.
Therefore, if two powers have the same base, their exponents must be equal:
\[
\begin{aligned}
a^{f(x)}=a^{g(x)}
\quad\Longrightarrow\quad
f(x)=g(x).
\end{aligned}
\]
This is often the fastest method.
3. Same-base example
Solve:
\[
\begin{aligned}
3^{2x-1}=27.
\end{aligned}
\]
Rewrite \(27\) as a power of \(3\):
\[
\begin{aligned}
27=3^3.
\end{aligned}
\]
Then:
\[
\begin{aligned}
3^{2x-1}&=3^3.
\end{aligned}
\]
Equate exponents:
\[
\begin{aligned}
2x-1&=3,\\
2x&=4,\\
x&=2.
\end{aligned}
\]
Check:
\[
\begin{aligned}
3^{2(2)-1}
&=
3^3\\
&=
27.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=2}.
\end{aligned}
\]
4. Logarithm method
When the right-hand side is not an obvious power of the base, use logarithms.
Starting from:
\[
\begin{aligned}
a^{kx+m}=b,
\end{aligned}
\]
take natural logarithms on both sides:
\[
\begin{aligned}
\ln\left(a^{kx+m}\right)&=\ln(b).
\end{aligned}
\]
Use the power rule:
\[
\begin{aligned}
(kx+m)\ln(a)&=\ln(b).
\end{aligned}
\]
Then solve for \(x\):
\[
\begin{aligned}
kx+m&=\frac{\ln(b)}{\ln(a)},\\
kx&=\frac{\ln(b)}{\ln(a)}-m,\\
x&=\frac{\ln(b)/\ln(a)-m}{k}.
\end{aligned}
\]
This method requires \(a>0\), \(a\ne1\), and \(b>0\).
5. Natural exponential equations
The base \(e\) appears very often:
\[
\begin{aligned}
e^{kx+m}=b.
\end{aligned}
\]
Since \(\ln(e)=1\), the logarithm step simplifies:
\[
\begin{aligned}
kx+m&=\ln(b),\\
x&=\frac{\ln(b)-m}{k}.
\end{aligned}
\]
6. Equations with different bases
If the bases are different, logarithms can still help. For example:
\[
\begin{aligned}
2^{x+1}=3^{x-2}.
\end{aligned}
\]
Take natural logarithms:
\[
\begin{aligned}
(x+1)\ln(2)&=(x-2)\ln(3).
\end{aligned}
\]
This becomes a linear equation in \(x\). Not every different-base equation is solved symbolically by the calculator,
so the graph and numerical verification are useful fallback tools.
7. Lambert W type equations
Some equations contain both an exponential expression and a linear expression in \(x\), such as:
\[
\begin{aligned}
e^{kx+m}=cx.
\end{aligned}
\]
These are not solved by ordinary logarithms alone because \(x\) appears both inside and outside the exponential.
They can be rearranged into a Lambert \(W\)-function form:
\[
\begin{aligned}
x=-\frac{1}{k}W\left(\frac{-ke^m}{c}\right).
\end{aligned}
\]
The Lambert \(W\)-function is defined by:
\[
\begin{aligned}
W(z)e^{W(z)}=z.
\end{aligned}
\]
When the argument is between \(-1/e\) and \(0\), two real branches may exist, so two real solutions may occur.
8. Graph interpretation
An exponential equation:
\[
\begin{aligned}
\text{LHS}(x)=\text{RHS}(x)
\end{aligned}
\]
can be graphed as two curves:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Real solutions occur where the two curves intersect. A graph helps locate solutions,
but the final answer should still be checked by substitution.
9. Numerical verification
After a candidate value is found, substitute it into the original equation:
\[
\begin{aligned}
\text{LHS}(x_{\text{candidate}})
\stackrel{?}{=}
\text{RHS}(x_{\text{candidate}}).
\end{aligned}
\]
The calculator compares the two sides within a numerical tolerance. This catches invalid results,
graph-only false positives, and unsupported symbolic cases.
10. Formula summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
11. Common mistakes
- Equating exponents when the bases are not the same.
- Taking logarithms of a nonpositive number in a real-number problem.
- Forgetting that \(a^x\) must have \(a>0\) and \(a\ne1\) in the usual real exponential model.
- Rounding too early during logarithm calculations.
- Assuming a graph intersection is valid without numerical substitution.
- Missing a second real solution in Lambert \(W\)-type equations.
Key idea: use the same-base rule when possible; otherwise use logarithms, and always verify the final value.
