What Is a Logarithm?

Math Talk format. Students solve mentally — no paper, no calculator. Reveal one problem at a time and keep prior work visible, or display all four at once for a faster pace. Either way, the goal is to surface the strategies, not just the answers.

Answers: (1) \(x = 2\) — \(4 \cdot 4 = 16\). (2) \(x = 3\) — \(5 \cdot 5 \cdot 5 = 125\). (3) \(x = 6\) — counting powers of 2: 2, 4, 8, 16, 32, 64. (4) \(x\) is between 2 and 3, closer to 3 (\(\approx 2.7\)) — \(10^{2} = 100\) is too small, \(10^{3} = 1000\) is too big. Challenge: (5) \(x = 0\) — anything to the zero power is 1. (6) \(x = -2\) — rewrite \(\tfrac{1}{100}\) as \(10^{-2}\).

The pivot. Problems 1–3 have exact integer exponents. Problem 4 is trapped between integer powers — that “between” intuition is the seed for the entire lesson. Do not resolve \(10^{x} = 500\) numerically here; leave the tension. It returns in the synthesis (Slide 3) and pays off when the log table appears in Activity 1.

Listen for: language like “the exponent that turns 10 into 500,” “a power of 10 between 2 and 3,” or “a decimal exponent.” That phrasing is the bridge to naming the logarithm in Activity 2.

The pivot moment of the warm-up. The first three problems are exact; the fourth is trapped between integer exponents. Today's whole lesson is about naming that trapped exponent.

Do not resolve \(10^{x} = 500\) numerically here. Leave the tension. The table on Slide 5 will give the value (\(x \approx 2.699\)); the word logarithm lands in Activity 2 on Slides 7–8. If a student already says “logarithm,” affirm but do not pivot — we want every student to feel the need for the name before we give it.

Listen for: students phrasing the answer as “the exponent that makes 10 give 500.” That sentence is the definition of a logarithm — capture it in a Collect-and-Display so you can point back to it later.

Launch (~3 min). Use Collect and Display: capture the words students use as they try to describe the right column — “exponent,” “power of 10,” “decimal exponent,” “approximation,” “in between.” Post these on the board for the rest of the activity.

Do not introduce the word logarithm yet — that lands in Activity 2 (Slides 7–8). The point of this launch is to set up the table as the thing that answers what the warm-up could not: it names the exponent trapped between 2 and 3.

Hold the tension. If a student already knows the word “logarithm,” thank them and ask them to hold it until Activity 2 so the class can discover the structure first.

Standards: HSF-BF.B.4, HSF-LE.A.4.

Pay off the warm-up cliffhanger: \(10^{x} = 500\) has solution \(x \approx 2.6990\) — the table told us, and the calculator confirms (10^(2.6990) returns about 500). Surface the structural insight from work time: multiplying the input by 10 adds exactly 1 to the exponent (compare row 9 vs. row 90). That's the seed of the log-product property — the “multiplication-into-addition” trick from the Big Question. Don't name logarithm yet — that lands on the next MASL slide. Leave the tension: we now have a number; what do we call it?

Pronounce slowly: “log base b of x.” Stress that the logarithm is simply a name for the exponent students were already reading out of the table in Activity 1 — the right-hand column of that table is, literally, a list of logarithms base 10. The biconditional arrow (\(\Longleftrightarrow\)) carries the entire lesson: every logarithmic equation has a partner exponential equation that says the same thing, and we can move between them freely. Do not compute a logarithm numerically here — that’s Activity 2. This slide is the naming moment. Pair immediately with the How To on Slide 8 (base-10 special case + rewriting procedure). No ACB — MASL slides are discussion/reference only.

Leave this slide on screen during Activity 2 work time. Component-part MASL is allowed here because Slide 7 introduced the general form \(\log_{b}(x) = y \iff b^{y} = x\) first. Pronounce the Math card slowly: "log base ten of x equals y." Stress: the logarithm is the exponent — the base of the log matches the base of the power, the value inside the log is what the power produces, and the answer is the exponent that makes the equation true. The How To is the procedural anchor for Activity 2 (Slide 9) and the cool-down (Slide 12). When you reach Step 5, link it to the calculator: the calculator's log button means base-10 by convention, which is why the subscript can be dropped.

Reinforce the biconditional structure: every log statement is the same as an exponential statement, just written differently. The arrow \(\iff\) from Slide 7 is the central object of the lesson. Tomorrow students will use this rewriting move to solve equations like \(10^{x} = 500\) exactly — naming the exponent is the answer.

Lesson Synthesis (~3 min). SWBAT belongs here, not on the title slide — read the “I Can” statements aloud after the second discussion point lands, so students hear the goals after they have done the thinking.

Tease tomorrow (Lesson 10): we will use the rewriting move to solve equations like \(10^{x} = 500\) for an exact log expression. Students do not need to evaluate that log to a decimal — naming the exponent is the answer. The log-product property \(\log(a \cdot b) = \log(a) + \log(b)\) is just a tease today; it is formally taught later in the unit, but planting it now ties the deck's Napier / slide-rule narrative back to today's definition.

Lesson Summary slide. Read aloud or have a student read. Closing tease: tomorrow, instead of naming the exponent, we will start solving exponential equations whose answers are logarithms — and in the next section, we will learn properties (\(\log(a \cdot b) = \log(a) + \log(b)\)) that explain the 300-year-old multiplication-into-addition trick from the Big Question.