Exponential Equations — Solving Methods, Formulas & Examples
Exponential Equations — Solving Methods, Formulas & Examples
What is an Exponential Equation?
An exponential equation is an equation in which the variable appears as part of an exponent.
2x=322^x = 322x=32 is an exponential equation. x2=32x^2 = 32x2=32 is not — the variable is the base there, not the exponent. The position of the variable changes the entire toolkit you need to solve the equation.
When bacteria double every twenty minutes, when interest compounds on top of yesterday's interest, when a radioactive sample loses half its mass every TTT years — the equation that describes the moment is exponential. The variable is the time. The base is the growth rate. The answer tells you when.
Types of Exponential Equations
Competitor pages typically split exponential equations into three categories, and the split is useful because each category needs a different solving move.
| Type | Example | Solving move |
|---|---|---|
| Same base on both sides | 4x=424^x = 4^24x=42 | Set the exponents equal: x=2x = 2x=2 |
| Different bases, can be made the same | 4x=164^x = 164x=16 → 4x=424^x = 4^24x=42 | Rewrite, then set exponents equal |
| Different bases, cannot be made the same | 4x=154^x = 154x=15 | Take the logarithm of both sides |
The first two are mechanical once you see them. The third needs logarithms — and that is where most exponential equations actually live.
The Three Methods, at a Glance
| Method | When to use it | Key move |
|---|---|---|
| Same base | Both sides can be written with the same base | Set the exponents equal: af(x)=ag(x)⇒f(x)=g(x)a^{f(x)} = a^{g(x)} \Rightarrow f(x) = g(x)af(x)=ag(x)⇒f(x)=g(x) |
| Logarithms | Bases cannot be matched | Take log\loglog on both sides: log(ax)=logb⇒x=logbloga\log(a^x) = \log b \Rightarrow x = \dfrac{\log b}{\log a}log(ax)=logb⇒x=logalogb |
| Substitution | The equation has the same exponential expression twice | Let u=axu = a^xu=ax and solve a quadratic in uuu |
How Do You Solve Exponential Equations? Three Worked Examples
We will walk through three problems — Quick, Standard, and Stretch. The Standard example opens with the tempting wrong path so you can see exactly where it fails.
Quick example
Quick. Solve 2x=322^x = 322x=32.
32=25⟹2x=25⟹x=532 = 2^5 \implies 2^x = 2^5 \implies x = 532=25⟹2x=25⟹x=5
Final answer: x=5x = 5x=5.
The tempting shortcut that doesn't work
Standard. Solve 5⋅3x=455 \cdot 3^x = 455⋅3x=45.
Wrong path. A rusher sees the 555 on the left and treats the whole left side as one block:
5⋅3x=45⟹(5⋅3)x=45⟹15x=45⟹x≈1.45 \cdot 3^x = 45 \implies (5 \cdot 3)^x = 45 \implies 15^x = 45 \implies x \approx 1.45⋅3x=45⟹(5⋅3)x=45⟹15x=45⟹x≈1.4
That is wrong. The base is 333, not 151515. The 555 is a coefficient sitting in front of the exponential — it does not get folded into the base.
Correct path. Isolate the exponential first, then solve.
5⋅3x=45⟹3x=9⟹3x=32⟹x=25 \cdot 3^x = 45 \implies 3^x = 9 \implies 3^x = 3^2 \implies x = 25⋅3x=45⟹3x=9⟹3x=32⟹x=2
Final answer: x=2x = 2x=2.
Stretch example
Stretch. Solve 4x−6⋅2x+8=04^x - 6 \cdot 2^x + 8 = 04x−6⋅2x+8=0.
Notice that 4x=(22)x=(2x)24^x = (2^2)^x = (2^x)^24x=(22)x=(2x)2. Let u=2xu = 2^xu=2x. The equation becomes a quadratic:
u2−6u+8=0⟹(u−2)(u−4)=0⟹u=2 or u=4u^2 - 6u + 8 = 0 \implies (u - 2)(u - 4) = 0 \implies u = 2 \text{ or } u = 4u2−6u+8=0⟹(u−2)(u−4)=0⟹u=2 or u=4
Now back-substitute:
2x=2⟹x=1,2x=4⟹x=22^x = 2 \implies x = 1, \quad 2^x = 4 \implies x = 22x=2⟹x=1,2x=4⟹x=2
Final answer: x=1x = 1x=1 or x=2x = 2x=2.
A sanity check on x=2x = 2x=2: 42−6⋅22+8=16−24+8=04^2 - 6 \cdot 2^2 + 8 = 16 - 24 + 8 = 042−6⋅22+8=16−24+8=0. The equation holds.
The Three Solving Methods in Detail
Method 1 — Same base
If you can rewrite both sides with the same base, the property of equality for exponential functions kicks in: af(x)=ag(x)⇒f(x)=g(x)a^{f(x)} = a^{g(x)} \Rightarrow f(x) = g(x)af(x)=ag(x)⇒f(x)=g(x) (provided a>0a > 0a>0, a≠1a \neq 1a=1).
Common rewrites worth remembering: 4=224 = 2^24=22, 8=238 = 2^38=23, 16=2416 = 2^416=24, 25=5225 = 5^225=52, 27=3327 = 3^327=33, 81=34=9281 = 3^4 = 9^281=34=92, 125=53125 = 5^3125=53, 1000=1031000 = 10^31000=103.
Method 2 — Logarithms (when bases can't be matched)
Take the log\loglog of both sides. The base of the log does not matter, as long as you are consistent — natural log (ln\lnln) is the conventional choice in calculus, common log (log10\log{10}log10) in scientific work.
ax=b⟹log(ax)=logb⟹xloga=logb⟹x=logblogaa^x = b \implies \log(a^x) = \log b \implies x \log a = \log b \implies x = \frac{\log b}{\log a}ax=b⟹log(ax)=logb⟹xloga=logb⟹x=logalogb
Method 3 — Substitution (for quadratic-in-disguise equations)
When the same exponential expression shows up twice (often as axa^xax and a2xa^{2x}a2x), substitute u=axu = a^xu=ax and reduce the equation to a polynomial in uuu. After solving, back-substitute and finish.
Why do Exponential Equations Matter? The Reach Beyond the Classroom
Exponential equations are not classroom decoration. They show up wherever a quantity's rate of change is proportional to its current size.
Compound interest. A=P(1+r/n)ntA = P(1 + r/n)^{nt}A=P(1+r/n)nt — the formula every bank, lender, and pension fund runs on. Solve for ttt when AAA doubles, and you have the doubling time of the investment.
Radioactive decay. Carbon-14 dating uses N(t)=N0⋅e−λtN(t) = N_0 \cdot e^{-\lambda t}N(t)=N0⋅e−λt. Archaeologists solve for ttt to age organic material. The Shroud of Turin was carbon-dated in 1988 using this exact equation.
Earthquakes — the Richter scale. Each whole-number step on the Richter scale corresponds to a tenfold increase in measured amplitude. A magnitude 6 quake releases 10 times the energy of a magnitude 5.
Pandemic modelling. The early days of the 2020 COVID-19 outbreak followed N(t)≈N0⋅1.3tN(t) \approx N0 \cdot 1.3^tN(t)≈N0⋅1.3t for days ttt. Solving for the day the case count crossed a million was a one-step exponential equation.
Cooling — Newton's law. A cup of coffee at 90°C90°C90°C in a 25°C25°C25°C room reaches half its temperature gap with the room every few minutes, following T(t)=Troom+(T0−Troom)e−kT(t) = T_\text{room} + (T_0 - T_\text{room}) e^{-kt}T(t)=Troom+(T0−Troom)e−kt.
Where Students Lose the Mark
Three errors account for most of the marks lost on exponential equations.
Mistake 1: Forgetting to isolate the exponential first.
Where it slips in: When a coefficient sits in front of the exponential, students try to apply a same-base move without dividing out the coefficient.
Don't do this: 5⋅3x=45⟹15x=455 \cdot 3^x = 45 \implies 15^x = 455⋅3x=45⟹15x=45.
The correct way: Isolate first. 5⋅3x=45⟹3x=9⟹x=25 \cdot 3^x = 45 \implies 3^x = 9 \implies x = 25⋅3x=45⟹3x=9⟹x=2. The coefficient is a multiplier, not part of the base.
Mistake 2: Treating log(a+b)\log(a + b)log(a+b) as loga+logb\log a + \log bloga+logb.
Where it slips in: When the exponential equation has a sum inside the exponent or on one side, students reach for the log-product rule and apply it to a sum.
Don't do this: log(2x+5)=log2x+log5\log(2^x + 5) = \log 2^x + \log 5log(2x+5)=log2x+log5.
The correct way: log(ab)=loga+logb\log(ab) = \log a + \log blog(ab)=loga+logb is correct. log(a+b)\log(a + b)log(a+b) does not split — it has no clean simplification. If the equation has a sum, isolate the exponential before logging.
Mistake 3: Dropping the positivity check on the substitution.
Where it slips in: In Method 3, after substituting u=axu = a^xu=ax and solving the polynomial, students forget that ax>0a^x > 0ax>0 for any real xxx. Any negative uuu from the polynomial must be discarded.
Don't do this: u=−2⟹2x=−2⟹x=log2(−2)u = -2 \implies 2^x = -2 \implies x = \log_2(-2)u=−2⟹2x=−2⟹x=log2(−2) — undefined.
The correct way: When the polynomial gives u=−2u = -2u=−2 or u=4u = 4u=4, reject u=−2u = -2u=−2 (because 2x2^x2x is always positive) and keep only u=4⟹x=2u = 4 \implies x = 2u=4⟹x=2.
The Mathematicians Who Built the Exponential Toolkit
Two figures whose work made solving exponential equations possible.
John Napier (1550–1617, Scotland). Napier invented logarithms — published in 1614 as Mirifici Logarithmorum Canonis Descriptio — explicitly to turn multiplication into addition. Every "take the log of both sides" move in this article descends from Napier's tables.
Leonhard Euler (1707–1783, Switzerland). Euler introduced the constant e≈2.71828e \approx 2.71828e≈2.71828 as the natural base for the exponential function and proved the identity eiπ+1=0e^{i\pi} + 1 = 0. The natural log — ln\lnln — is the inverse of Euler's exponential, and most physical exponential decay/growth equations use eee as the base for that reason.
Conclusion
An exponential equation has the variable in the exponent, not the base.
Three solving methods cover every standard case: same base, logarithms, substitution.
Always isolate the exponential before applying same-base or log moves.
axa^xax is always positive — reject negative substitution results.
Real-world exponential equations govern compounding, decay, earthquakes, and epidemic growth.
A practical next step
Three problems to practise. If you stall on any of them, come back to the matching worked example above.
Solve 2x+1=322^{x+1} = 322x+1=32.
Solve 3x=503^x = 503x=50 to two decimal places.
Solve 9x−4⋅3x+3=09^x - 4 \cdot 3^x + 3 = 09x−4⋅3x+3=0.