Lesson 14

Solving Exponential Equations

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Problem 1

Solve each equation without using a calculator. Some solutions will need to be expressed using log notation.

  1. \(4 \boldcdot 10^x = 400,\!000\)
  2. \(10^{(n+1)} = 1\)
  3. \(10^{3n} = 1,\!000,\!000\)
  4. \(10^p = 725\)
  5. \(6 \boldcdot 10^t = 360\)

Solution

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Problem 2

Solve \(\frac14 \boldcdot 10^{(d+2)} = 0.25\). Show your reasoning.

Solution

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Problem 3

Write two equations—one in logarithmic form and one in exponential form—that represent the statement: “the natural logarithm of 10 is \(y\)”.

Solution

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Problem 4

Explain why \(\ln 1 = 0\).

Solution

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Problem 5

If \(\log_{10}(x) = 6\), what is the value of \(x\)? Explain how you know.

Solution

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(From Unit 4, Lesson 9.)

Problem 6

For each logarithmic equation, write an equivalent equation in exponential form.

  1. \(\log_2 16 = 4\)
  2. \(\log_3 9 = 2\)
  3. \(\log_5 5 = 1\)
  4. \(\log_{10} 20 = y\)
  5. \(\log_2 30 = y\)

Solution

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(From Unit 4, Lesson 10.)

Problem 7

The function \(f\) is given by \(f(x) = e^{0.07x}\).

  1. What is the continuous growth rate of \(f\)?
  2. By what factor does \(f\) grow when the input \(x\) increases by 1? 

Solution

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(From Unit 4, Lesson 13.)