Today's lesson gave us four moves for finding an unknown exponent. Below you'll find a quick worked example for each method, followed by 2 problems to solve using that method. Eight problems total — one method per section.
Print the 2-sided worksheet (top right) and write your answers there to hand in.
Method 1 — Repeat-multiply
Multiply the base by itself, counting as you go, until you hit the target. Best when the answer is a clean integer.
Worked example: Solve \(3^{\,x} = 81\)
Write each step in expanded form — the base multiplied by itself, one factor at a time. Read each line in plain English to see why the exponent is what it is:
\[\begin{aligned}
3 &= 3 && \text{So 3 by itself is } 3^{1} \\
3 \cdot 3 &= 9 && \text{So 3 times itself two times is } 3^{2} \\
3 \cdot 3 \cdot 3 &= 27 && \text{So 3 times itself three times is } 3^{3} \\
3 \cdot 3 \cdot 3 \cdot 3 &= 81 \,\checkmark && \text{So 3 times itself four times is } 3^{4}
\end{aligned}\]
The 4th multiplication hits the target, so \(x = 4\).
Solve \(2^{\,x} = 64\).
Solve \(5^{\,x} = 125\).
Method 2 — Calculator + decimals
Try decimal exponents on a calculator and adjust until the result matches the target. Best when you need ~2 decimal places of precision.
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 \(x \approx 3.32\).
\(4^{\,x} = 50\) (Solve for x, round to 2 decimals)
\(7^{\,x} = 100\) (Solve for x, round to 2 decimals)
Method 3 — Sandwich + estimate
Find the two 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: \(2.4\).
Solve \(3^{\,x} = 50\). (\(x\) is between which two integers? Then estimate \(x\) to 1 decimal.)
Solve \(5^{\,x} = 200\). (\(x\) is between which two integers? Then estimate \(x\) to 1 decimal.)
Method 4 — Graph and read off (Desmos)
Graph the exponential, draw the horizontal target line, and read the intersection's \(x\)-value.
Worked example: Solve \(5^{\,x} = 100\)
Open Desmos. Type in both equations: y = 5^x y = 100
Click the intersection of the curve and the line. Desmos shows the point \((2.861, 100)\), so \(x \approx 2.86\).
The blue curve \(y = 5^x\) meets the red target line \(y = 100\) at \(x \approx 2.86\).
\(2^{\,x} = 25\) (Solve for x, round to 2 decimals.)
Graph both: y = 2^x y = 25.
\(6^{\,x} = 1000\) (Solve for x, round to 2 decimals.)
Graph both: y = 6^x y = 1000.
Reflect — One More Question
Which method seemed the easiest to you, and why?
Try Writing
The method that seemed the easiest to me was (circle one)Repeat-multiply / Calculator + decimals / Sandwich + estimate / Graph and read off because __________.