A logarithmic equation is an equation containing one or more logarithms of expressions involving a variable.
The most important rule is that every logarithm must have a positive argument and a valid base.
Candidate answers must always be checked in the original equation.
1. What is a logarithmic equation?
Examples include:
\[
\begin{aligned}
\log_2(x+1)+\log_2(x-1)&=3,\\
\log_3(2x-1)&=4,\\
\ln(x)&=2,\\
\ln(x)&=0.5x.
\end{aligned}
\]
A logarithmic equation can often be solved by combining logarithms and then converting the logarithmic form into exponential form.
2. Domain restrictions
A logarithm is defined only when its argument is positive:
\[
\begin{aligned}
\log_b(A(x))\text{ is defined only if }A(x)>0.
\end{aligned}
\]
The base must also satisfy:
\[
\begin{aligned}
b>0,\qquad b\ne1.
\end{aligned}
\]
These restrictions must be applied before accepting a final solution.
3. Change of base
Any logarithm can be rewritten using natural logarithms:
\[
\begin{aligned}
\log_b(A)=\frac{\ln(A)}{\ln(b)}.
\end{aligned}
\]
This explains why logarithmic equations with different bases can still be compared numerically or transformed algebraically.
4. Basic logarithmic form
The equation:
\[
\begin{aligned}
\log_b(A)=k
\end{aligned}
\]
is equivalent to:
\[
\begin{aligned}
A=b^k.
\end{aligned}
\]
This conversion is called exponentiating both sides.
5. Product and quotient rules
When logarithms have the same base, they can be combined:
\[
\begin{aligned}
\log_b(M)+\log_b(N)&=\log_b(MN),\\
\log_b(M)-\log_b(N)&=\log_b\left(\frac{M}{N}\right).
\end{aligned}
\]
These laws are useful, but they do not remove the original domain restrictions.
6. Worked example
Solve:
\[
\begin{aligned}
\log_2(x+1)+\log_2(x-1)=3.
\end{aligned}
\]
First state the domain:
\[
\begin{aligned}
x+1&>0,\\
x-1&>0.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
x>1.
\end{aligned}
\]
Combine logs using the product rule:
\[
\begin{aligned}
\log_2\big((x+1)(x-1)\big)=3.
\end{aligned}
\]
Exponentiate:
\[
\begin{aligned}
(x+1)(x-1)&=2^3,\\
x^2-1&=8,\\
x^2&=9.
\end{aligned}
\]
The algebraic candidates are:
\[
\begin{aligned}
x=\pm3.
\end{aligned}
\]
Apply the domain \(x>1\). The candidate \(x=-3\) is invalid, while \(x=3\) is allowed.
Check:
\[
\begin{aligned}
\log_2(3+1)+\log_2(3-1)
&=
\log_2(4)+\log_2(2)\\
&=
2+1\\
&=
3.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=3}.
\end{aligned}
\]
7. Why extraneous solutions happen
Logarithmic equations often produce extra candidates after combining logs or exponentiating.
For example, in the worked example, \(x=-3\) satisfies the transformed equation \(x^2=9\),
but it does not satisfy the original domain because \(x-1<0\).
This is why the original logarithmic arguments must always be checked.
8. Difference of logarithms
Consider:
\[
\begin{aligned}
\log_2(x+3)-\log_2(x-1)=2.
\end{aligned}
\]
Domain:
\[
\begin{aligned}
x+3>0,\qquad x-1>0,
\end{aligned}
\]
so \(x>1\). Combine logs:
\[
\begin{aligned}
\log_2\left(\frac{x+3}{x-1}\right)=2.
\end{aligned}
\]
Exponentiate:
\[
\begin{aligned}
\frac{x+3}{x-1}=4.
\end{aligned}
\]
Then solve and check the domain.
9. Natural logarithm equations
The natural logarithm uses base \(e\):
\[
\begin{aligned}
\ln(x)=k
\quad\Longleftrightarrow\quad
x=e^k.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\ln(x)=2
\quad\Longrightarrow\quad
x=e^2.
\end{aligned}
\]
10. Hybrid logarithmic equations
Some equations mix a logarithm with a linear expression:
\[
\begin{aligned}
\ln(x)=ax+b.
\end{aligned}
\]
These equations usually cannot be solved by elementary algebra alone.
They may require the Lambert \(W\)-function or numerical methods.
The Lambert \(W\)-function is defined by:
\[
\begin{aligned}
W(z)e^{W(z)}=z.
\end{aligned}
\]
Hybrid equations can have no real solution, one real solution, or two real solutions depending on the graph.
11. Graph interpretation
A logarithmic equation:
\[
\begin{aligned}
\text{LHS}(x)=\text{RHS}(x)
\end{aligned}
\]
can be graphed as two curves:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Solutions occur where the two curves intersect and all logarithmic arguments are positive.
Domain shading on the graph helps show where the original equation is not defined.
12. Formula summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
13. Common mistakes
- Forgetting that each log argument must be positive.
- Using log laws with different bases without changing base first.
- Keeping an algebraic candidate that violates the original domain.
- Assuming \(\log_b(M+N)=\log_b(M)+\log_b(N)\), which is false.
- Exponentiating correctly but forgetting to verify in the original equation.
- Rounding too early when using logarithms or numerical methods.
Key idea: solve logarithmic equations by combining valid logs, exponentiating, and checking every candidate in the original domain.
