Solving Exponential Equations

An **exponential equation** is an equation where the unknown is in the **exponent**, for example $2^x = 32$ or $3^{x+1} = 81$. The key idea is the **same-base rule**: if two powers with the same base are equal, their exponents are equal. $$b^m = b^n \implies m = n \quad (b > 0,\ b \neq 1)$$ **The same-base method:** 1. **Isolate the power**: if a number multiplies the power, divide both sides by it ($3 \cdot 2^x = 48 \implies 2^x = 16$). 2. **Write both sides as powers of the same base** ($16 = 2^4$, $8 = 2^3$, $4 = 2^2$). 3. **Set the exponents equal** and solve the simpler equation. 4. **Check** by substituting the solution back into the original equation. Useful rewrites: - Fractions become **negative exponents**: $\frac{1}{27} = 3^{-3}$ - Roots become **fractional exponents**: $\sqrt{2} = 2^{\frac{1}{2}}$ - Power of a power: $4^x = (2^2)^x = 2^{2x}$ When the two sides cannot be written with the same base (like $2^x = 20$), you can **estimate** the solution with a table of powers: $2^4 = 16$ and $2^5 = 32$, so $x$ lies between $4$ and $5$. **Logarithms**, a later topic, give the exact value.

A Direct Same-Base Equation

Solve $2^x = 32$.

Write 32 as a power of 2: $32 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5$ = $2^x = 2^5$

Set the exponents equal: Same base $2$, so the exponents must match = $x = 5$

Check: $2^5 = 32$ ✓ = The solution is correct

Answer: $x = 5$

An Expression in the Exponent

Solve $3^{x+1} = 81$.

Write 81 as a power of 3: $81 = 3^4$ = $3^{x+1} = 3^4$

Set the exponents equal: $x + 1 = 4$ = A linear equation

Solve for $x$: $x = 4 - 1$ = $x = 3$

Check: $3^{3+1} = 3^4 = 81$ ✓ = The solution is correct

Answer: $x = 3$

Rewriting Both Sides

Solve $4^x = 8$.

Find a common base: $4 = 2^2$ and $8 = 2^3$, so both are powers of $2$ = Common base: $2$

Rewrite the left side: $4^x = (2^2)^x = 2^{2x}$ = $2^{2x} = 2^3$

Set the exponents equal: $2x = 3$ = $x = \frac{3}{2}$

Check: $4^{\frac{3}{2}} = (\sqrt{4})^3 = 2^3 = 8$ ✓ = The solution is correct

Answer: $x = \frac{3}{2} = 1.5$

A Fraction on the Right Side

Solve $9^x = \frac{1}{27}$.

Rewrite the left side with base 3: $9^x = (3^2)^x = 3^{2x}$ = $3^{2x}$

Rewrite the fraction as a negative exponent: $\frac{1}{27} = \frac{1}{3^3} = 3^{-3}$ = $3^{2x} = 3^{-3}$

Set the exponents equal: $2x = -3$ = $x = -\frac{3}{2}$

Check: $9^{-\frac{3}{2}} = \frac{1}{9^{\frac{3}{2}}} = \frac{1}{(\sqrt{9})^3} = \frac{1}{27}$ ✓ = The solution is correct

Answer: $x = -\frac{3}{2} = -1.5$

Estimating a Solution with a Table

Solve $5 \cdot 2^x = 100$. If the solution is not a whole number, find the two consecutive integers it lies between.

Isolate the power: Divide both sides by $5$: $2^x = \frac{100}{5}$ = $2^x = 20$

Try to find a common base: $20 = 2^2 \cdot 5$ is not a power of $2$ = The same-base method does not work

Make a table of powers of 2: $2^3 = 8$, $2^4 = 16$, $2^5 = 32$ = $16 < 20 < 32$

Locate the solution: $2^4 < 2^x < 2^5$, and a larger exponent always gives a larger power of $2$ = $4 < x < 5$

Refine with a calculator: $2^{4.3} \approx 19.7$ and $2^{4.4} \approx 21.1$ = $x \approx 4.3$

Answer: $x$ lies between $4$ and $5$ (about $4.32$). Logarithms, a later topic, give the exact value.

Exponential equations answer the question **"when?"** for anything that grows or shrinks by a constant factor: - **Biology**: When will a bacteria colony that doubles every hour reach 4800 cells? - **Finance**: How many years until an investment doubles or triples? - **Technology**: How many doublings of storage are needed to hold a given amount of data? - **Science**: How long until a substance drops to a safe level? In the previous lessons you calculated the amount for a given time. Here you turn the question around and find the **time (the exponent)** for a given amount. The same-base method also trains you to see numbers like $8$, $81$ and $\frac{1}{27}$ as powers, a skill you will use again with logarithms.

Mistake: Dividing by the base: $2^x = 32 \implies x = \frac{32}{2} = 16$

Why: $2^x$ means $2$ multiplied by itself $x$ times, not $2$ times $x$. Dividing undoes multiplication, not exponentiation.

Correct: Write $32$ as a power of $2$: $32 = 2^5$, so $x = 5$. Check: $2^{16} = 65536 \neq 32$.

Mistake: Multiplying the factor into the base: $3 \cdot 2^x = 48 \implies 6^x = 48$

Why: The exponent belongs only to the $2$. By the order of operations, the power is calculated first and then multiplied by $3$.

Correct: Divide first: $2^x = 16 = 2^4$, so $x = 4$. Check: $3 \cdot 2^4 = 3 \cdot 16 = 48$ ✓

Mistake: Setting exponents equal when the bases differ: $4^x = 2^3 \implies x = 3$

Why: The same-base rule only works when the bases are really the same. $4^3 = 64$, not $8$.

Correct: First rewrite $4^x = 2^{2x}$, then $2x = 3$, so $x = \frac{3}{2}$.

Mistake: Losing the minus sign: $9^x = \frac{1}{27} \implies x = \frac{3}{2}$

Why: A number smaller than $1$ needs a negative exponent when the base is bigger than $1$: $\frac{1}{27} = 3^{-3}$.

Correct: $3^{2x} = 3^{-3}$, so $x = -\frac{3}{2}$. Quick test: $9^{\frac{3}{2}} = 27$, which is far bigger than $\frac{1}{27}$.

Bacteria Colonies

A colony that doubles every hour follows $N = N_0 \cdot 2^t$. Solving for $t$ tells a lab when the colony reaches a target size.

A colony of 100 cells doubles every hour. It reaches 800 cells when $100 \cdot 2^t = 800$, so $2^t = 8 = 2^3$ and $t = 3$ hours.

Folding Paper

Each fold doubles the thickness of a sheet of paper, so the thickness after $n$ folds is $T = T_0 \cdot 2^n$.

A sheet 0.1 mm thick is 0.8 mm thick after 3 folds, because $0.1 \cdot 2^3 = 0.8$.

Viral Videos

Views that double every day grow like $V = V_0 \cdot 2^d$. When the target is not a power of 2, a table of powers shows the day it is passed.

A video with 1000 views that doubles daily has $1000 \cdot 2^5 = 32000$ views after 5 days and $1000 \cdot 2^6 = 64000$ after 6 days.

An exponential equation has the unknown in the exponent, like $2^x = 32$

Same-base rule: if $b^m = b^n$ (with $b > 0$, $b \neq 1$), then $m = n$

Isolate the power first: divide away any factor in front, as in $3 \cdot 2^x = 48 \implies 2^x = 16$

Rewrite with a common base using $(b^m)^x = b^{mx}$, negative exponents for fractions ($\frac{1}{27} = 3^{-3}$) and fractional exponents for roots

Always check the solution by substituting it back into the original equation

If no common base exists, a table of powers shows between which integers the solution lies; logarithms give the exact value

Q: What if I cannot write both sides with the same base?

A: Then the same-base method does not apply. You can still estimate: make a table of powers and find where the right side fits, for example $2^4 = 16 < 20 < 32 = 2^5$, so the solution of $2^x = 20$ is between $4$ and $5$. Logarithms, which come later in the course, give the exact answer.

Q: Why must the base not be 1?

A: Because $1^x = 1$ for every $x$. The equation $1^x = 1^5$ is true for all values of $x$, so you cannot conclude that $x = 5$. With any positive base other than $1$, different exponents always give different results.

Q: Can an exponential equation have no solution?

A: Yes. A power with a positive base is always positive, so $2^x = -4$ and $2^x = 0$ have no solution. Before you start, check that the isolated power equals a positive number.

Q: Can there be two solutions?

A: Yes, when the exponent contains $x^2$. For example $2^{x^2} = 2^4$ gives $x^2 = 4$, so $x = 2$ or $x = -2$. Check both in the original equation.

Exit Presentation

Solving Exponential Equations

1 / 14

Pre-Calculus / Exponential Functions

Solving Exponential Equations

Solve equations with the variable in the exponent by writing both sides as powers of the same base, and estimate solutions that are not whole numbers.

Advanced25 min

Use arrow keys or spacebar to navigate