Exponential Equations

Exponential equations place the unknown in an exponent and can be solved with common bases, graphs, or logarithms.

Equal input steps create multiplicative output change. An increasing curve stays above a horizontal asymptote, grows slowly at first, and then rises rapidly. Three marked points show a constant multiplication factor rather than a constant difference.

To solve an exponential equation, isolate the exponential expression and choose a method that respects its structure.

Recognizing the equation type

An exponential equation contains a variable in an exponent, such as

3^{x+1}=81.

This differs from $x^3=81$, where the variable is the base. Ordinary polynomial techniques do not move a variable down from an exponent; use equivalent bases, graphs, or logarithms.

Method 1: write a common base

If both sides can be expressed with the same positive base other than $1$, equate the exponents because the exponential function is one-to-one.

For example,

3^{x+1}=81=3^4,

$$ x+1=4,qquad x=3. $$

This method gives exact answers quickly when powers are recognizable.

Worked example with exponent laws

Substitution in the original equation confirms the equality.

Method 2: graph the intersection

An equation $f(x)=g(x)$ can be solved by graphing $y=f(x)$ and $y=g(x)$ and finding their intersection coordinates. This is useful when no easy common base exists or when a course expects technology-assisted solutions.

Graphing provides an estimate, so set a sensible viewing window and verify the reported $x$-value by substitution. Multiple intersections are possible in equations that combine exponential and other functions.

Method 3: use logarithms

Logarithms undo exponentiation. For

$$ 2^x=7, $$

take a logarithm of both sides:

\log(2^x)=\log7.

Bring down the exponent:

x\log2=\log7,

x=\frac{\log7}{\log2}\approx2.807.

Natural logarithms work equally well: $x=\ln7/\ln2$.

Common mistakes

Equating exponents when bases differ. First rewrite with a genuinely common base.

Distributing an exponent over addition. $(a+b)^x$ is not $a^x+b^x$.

Taking a logarithm of only one term. Apply an operation to both complete sides of the equation.

Rounding too early. It can shift a predicted time or threshold.

Ignoring the range. A positive exponential cannot equal a negative target before a vertical shift is considered.