Equal powers and solving for the exponent

If $2^x = 2^5$, what is $x$? It must be $5$. This sim­ple idea, law 11, lets you solve equa­tions where the unknown sits in the expo­nent, such as $2^{x+5} = 16$. The trick is to write both sides as pow­ers of the same base, then com­pare the expo­nents.

Law 11: equal pow­ers#

Law 11. If $a^m = a^n$, then $m = n$, where $a \neq -1, 0, 1$.

If two pow­ers of the same base are equal, their expo­nents are equal.

Why not the bases -1, 0 and 1?#

For these three bases, dif­fer­ent expo­nents can give the same answer.

• $1^4 = 1^7$, because both are $1$. But $4 \neq 7$.
• $0^2 = 0^3$, because both are $0$. But $2 \neq 3$.
• $(-1)^2 = (-1)^4$, because both are $1$. But $2 \neq 4$.

So the law only works for other bases.

Exam­ple 1#

$$2^x = 2^5 \Rightarrow x = 5 \qquad 7^x = 7^{29} \Rightarrow x = 29$$

Solv­ing for the expo­nent#

1. Write both sides as pow­ers of the same base. Use repeated divi­sion to turn a num­ber into a power.
2. Set the expo­nents equal (law 11).
3. Solve the sim­ple equa­tion you get.

Exam­ple 2#

Solve $17^{x-2} = 17^3$.

$$\begin{aligned} 17^{x-2} &= 17^3 \\ \Rightarrow x - 2 &= 3 \\ \Rightarrow x &= 3 + 2 = 5 \end{aligned}$$

Exam­ple 3#

Solve $2^{x+14} = 2^{29}$.

$$\begin{aligned} x + 14 &= 29 \\ \Rightarrow x &= 29 - 14 = 15 \end{aligned}$$

Exam­ple 4#

If $2^{x+5} = 16$, find $x$.

First write $16$ as a power of $2$. Divid­ing by $2$: $16$, $8$, $4$, $2$, $1$, four times. So $16 = 2^4$.

$$\begin{aligned} 2^{x+5} &= 16 \\ \Rightarrow 2^{x+5} &= 2^4 \\ \Rightarrow x + 5 &= 4 \\ \Rightarrow x &= 4 - 5 = -1 \end{aligned}$$

Exam­ple 5#

If $2^{2x+3} = 2^7$, find $x$.

$$\displaystyle \begin{aligned} 2x + 3 &= 7 \\ \Rightarrow 2x &= 7 - 3 = 4 \\ \Rightarrow x &= \frac{4}{2} = 2 \end{aligned}$$

Exam­ple 6#

If $5^{2x+5} = 625$, find $x$.

Divid­ing $625$ by $5$: $625$, $125$, $25$, $5$, $1$, four times. So $625 = 5^4$.

$$\displaystyle \begin{aligned} 5^{2x+5} &= 5^4 \\ \Rightarrow 2x + 5 &= 4 \\ \Rightarrow 2x &= 4 - 5 = -1 \\ \Rightarrow x &= -\frac{1}{2} \end{aligned}$$

The answer can be a frac­tion.

Exam­ple 7#

If $7^{5x-6} = 343$, find $x$.

$343 = 7 \times 7 \times 7 = 7^3$.

$$\displaystyle \begin{aligned} 5x - 6 &= 3 \\ \Rightarrow 5x &= 3 + 6 = 9 \\ \Rightarrow x &= \frac{9}{5} \end{aligned}$$

Exam­ple 8#

If $2^{2n} = 4096$, find $n$.

Divid­ing $4096$ by $2$ again and again takes twelve steps, so $4096 = 2^{12}$.

$$\begin{aligned} 2^{2n} &= 2^{12} \\ \Rightarrow 2n &= 12 \\ \Rightarrow n &= 6 \end{aligned}$$

Exam­ple 9: use another law first#

If $(-3)^{m+1} \times (-3)^5 = (-3)^7$, find $m$.

Join the left side with Law 1, then com­pare.

$$\begin{aligned} (-3)^{m+1+5} &= (-3)^7 \\ \Rightarrow (-3)^{m+6} &= (-3)^7 \\ \Rightarrow m + 6 &= 7 \\ \Rightarrow m &= 1 \end{aligned}$$

The base $-3$ is allowed. Only $-1$, $0$ and $1$ are not.

Exam­ple 10#

If $5^m \div 5^{-3} = 5^5$, find $m$.

Move $5^{-3}$ to the other side, where it mul­ti­plies.

$$\begin{aligned} 5^m &= 5^5 \times 5^{-3} \\ \Rightarrow 5^m &= 5^{5 + (-3)} = 5^2 \\ \Rightarrow m &= 2 \end{aligned}$$

Exam­ple 11: a frac­tion on the right#

If $\displaystyle 5^{3m+1} = \frac{1}{125}$, find $m$.

$125 = 5^3$, so $\displaystyle \frac{1}{125} = \frac{1}{5^3} = 5^{-3}$ by Law 7.

$$\displaystyle \begin{aligned} 5^{3m+1} &= 5^{-3} \\ \Rightarrow 3m + 1 &= -3 \\ \Rightarrow 3m &= -3 - 1 = -4 \\ \Rightarrow m &= -\frac{4}{3} \end{aligned}$$

Remem­ber. Same base on both sides, then the expo­nents are equal. Turn num­bers into pow­ers by repeated divi­sion, and frac­tions into neg­a­tive pow­ers.

Com­mon mis­takes#

• Com­par­ing expo­nents when the bases are dif­fer­ent. First make the bases the same.
• Writ­ing $16$ as $2^8$ or $4^4$. Check by divid­ing: $16 = 2^4$.
• Sign slips in the last step, as in $4 - 5 = -1$.
• For­get­ting that $\displaystyle \frac{1}{125}$ is a neg­a­tive power: $5^{-3}$.

Prac­tice#

1. If $3^{x-1} = 27$, find $x$.
2. If $7^{x+2} = 343$, find $x$.
3. If $3^{x+2} = 81$, find $x$.
4. If $10^{x+1} = 1000$, find $x$.
5. If $2^{3x-1} = 32$, find $x$.
6. If $4^{2x+7} = 64$, find $x$.
7. If $6^{2x-1} = 216$, find $x$.
8. If $5^{x-3} \times 5^3 = 625$, find $x$.
9. If $\displaystyle 2^{x-2} = \frac{1}{8}$, find $x$.
10. If $9^x = 729$, find $x$.

Answers#