Thirty-Six

Unit 5 · FunctionsF5 · Exponentials & Logs · Guidance

Logarithms & Exponential Equations

Use logarithms to expose exponents and solve models with unknown time.

  • 30 min
  • Unlocks: Burn Scheduler
  • Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

logarithm
the exponent needed to produce a given positive number

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Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

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The big idea

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The decision that matters

If the bases do not match, isolate the exponential and take logs of both sides. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

If the bases do not match, isolate the exponential and take logs of both sides. Keep exact values through the setup; round only when the stem asks.
Core relationship
ax=b⟺x=ln⁡bln⁡aa^x=b\Longleftrightarrow x=\frac{\ln b}{\ln a}

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Logarithms give an exact setup when the variable is in an exponent. Desmos can instead find the intersection of the exponential curve and target value; use the precision requested in the question.

Explore in Desmos

Solve 3x+1=503^{x+1}=50 by intersection

Click the intersection; use logarithms when an exact symbolic solution is needed.

The calculator is live: change anything and see what happens.

Solve 3^{x+1}=50 by intersection

Degree mode

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Trap

The main trap

Logarithms do not distribute across addition: ln⁡(a+b)\ln(a+b) is not ln⁡a+ln⁡b\ln a+\ln b.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1

Problem

Solve 3x+1=503^{x+1}=50 to the nearest thousandth.
Take natural logs: (x+1)ln⁡3=ln⁡50(x+1)\ln3=\ln50.

After taking natural logs, which equation correctly uses the power rule?

x≈2.561x\approx2.561.

Why is division by ln⁡3\ln3 valid?

Worked example

Example 2: ACT twist

Problem

Solve 500(0.8)t=200500(0.8)^t=200 to the nearest thousandth.
Divide by 500: 0.8t=0.40.8^t=0.4.

Which first operation isolates the exponential factor?

t≈4.106t\approx4.106.

Why must the coefficient 500 be removed before applying logarithms?

Faded example · you finish it

Example 3: You finish it

Problem

Solve 2x=132^x=13 to the nearest tenth.
x=ln⁡13ln⁡2x=\frac{\ln13}{\ln2}

Complete the calculation.

Desmos way

Compare the logarithmic and graphical solutions

Click the intersection; use logarithms when an exact symbolic solution is needed.

Compare the logarithmic and graphical solutions

Degree mode

Press Escape to leave the calculator (twice if a menu is open).

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Spot the mistake

Where did it go wrong?

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.