Algebra 2 · Unit 5 · Lesson 8 · Strategy Practice

Four methods for finding an unknown exponent. Read each worked example, then use that method to solve both problems in the section. Show work in the dashed boxes; write your answer on the line. 8 problems total + 1 reflection.

Method 1 — Repeat-multiply

Multiply the base by itself, one factor at a time. Best when the answer is a clean integer.

Worked example: Solve \(3^{\,x} = 81\) Write each step in expanded form. Stop when you hit the target: \[\begin{aligned} 3 &= 3 && \text{So 3 by itself is } 3^{1} \\ 3 \cdot 3 &= 9 && \text{So 3 times itself two times is } 3^{2} \\ 3 \cdot 3 \cdot 3 &= 27 && \text{So 3 times itself three times is } 3^{3} \\ 3 \cdot 3 \cdot 3 \cdot 3 &= 81 \,\checkmark && \text{So 3 times itself four times is } 3^{4} \end{aligned}\] The 4th multiplication hits the target, so \(x = 4\).

Your turn:

1. Solve \(2^{\,x} = 64\). \(x = \)

2. Solve \(5^{\,x} = 125\). \(x = \)

Method 2 — Calculator + decimals

Try decimal exponents on a calculator and adjust until the result matches the target. Round to 2 decimals.

Worked example: Solve \(2^{\,x} = 10\) Since \(2^{3} = 8\) (too small) and \(2^{4} = 16\) (too big), then we know \(x\) is between 3 and 4.
Try 2^3.3 → ≈ 9.85 (still under). Try 2^3.4 → ≈ 10.56 (too big). Try 2^3.35 → ≈ 10.20 (still too big). Try 2^3.32 → ≈ 9.99 (rounds to 10, target reached). So \(x \approx 3.32\).

Your turn:

3. \(4^{\,x} = 50\) (Solve for x, round to 2 decimals) \(x \approx \)

4. \(7^{\,x} = 100\) (Solve for x, round to 2 decimals) \(x \approx \)

Page 1 of 2 · Algebra 2 · Unit 5 · Lesson 8 · Strategy Practice

Method 3 — Sandwich + estimate

Find consecutive integers \(n\) and \(n+1\) with \(b^{n} < y < b^{n+1}\). Estimate where between them the answer lies.

Worked example: Solve \(4^{\,x} = 30\) \(4^{2} = 16\) and \(4^{3} = 64\), so \(2 < x < 3\).
Distance from 30 to 16 is 14; distance from 30 to 64 is 34. The target is closer to 16, so \(x\) is closer to 2 than to 3.
Estimate to one decimal: \(2.4\).

Your turn (give integers \(x\) is between, then estimate to 1 decimal):

5. Solve \(3^{\,x} = 50\). \(x\) is between and ; \(x \approx \)

6. Solve \(5^{\,x} = 200\). \(x\) is between and ; \(x \approx \)

Method 4 — Graph and read off (Desmos)

Graph the exponential, draw the horizontal target line, click the intersection. Round to 2 decimals.

Worked example: Solve \(5^{\,x} = 100\) Open Desmos. Type in both equations:
y = 5^x
y = 100
Click the intersection. Desmos shows \((2.861, 100)\), so \(x \approx 2.86\).

Your turn (Desmos at desmos.com/calculator):

7. \(2^{\,x} = 25\) (Solve for x, round to 2 decimals.)
Graph both:
y = 2^x
y = 25. \(x \approx \)

8. \(6^{\,x} = 1000\) (Solve for x, round to 2 decimals.)
Graph both:
y = 6^x
y = 1000. \(x \approx \)

Reflect — Which method seemed easiest?

The method that seemed the easiest to me was (circle one) Repeat-multiply / Calculator + decimals / Sandwich + estimate / Graph and read off because

Page 2 of 2 · Algebra 2 · Unit 5 · Lesson 8 · Strategy Practice