Solve exponential equations in any base with logarithms

About this lesson

Precalculus › Exponential and logarithmic functions

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Solve exponential equations in any base with logarithms

Warm-up

Three quick questions from earlier lessons.

Discover

You can solve 2x=82^x = 8 in your head. What about 2x=72^x = 7?

Which two whole numbers is the answer between? How could logarithms give it exactly?

Key idea

To solve a⋅bx=ca \cdot b^x = c: isolate the power, then take ln⁡\ln of both sides and use the power rule.

bx=k  ⟺  x=ln⁡kln⁡bb^x = k \iff x = \dfrac{\ln k}{\ln b} (for k>0k > 0)

Worked example

Solve 3⋅2x=213 \cdot 2^x = 21.

2x=72^x = 7
Divide both sides by 3 to isolate the power.
xln⁡2=ln⁡7x \ln 2 = \ln 7
Take ln of both sides; power rule.
x=ln⁡7ln⁡2≈2.81x = \dfrac{\ln 7}{\ln 2} \approx 2.81
Divide by ln 2.

Put the steps for solving a⋅bx=ca \cdot b^x = c in order.

  1. Use the power rule: xln⁡b=ln⁡cax \ln b = \ln \frac{c}{a}
  2. Divide by ln⁡b\ln b
  3. Take ln⁡\ln of both sides
  4. Divide both sides by aa

Your turn, guided

Your turn: solve 5x=125^x = 12. Fill in the missing line.

xln⁡5=ln⁡12x \ln 5 = \ln 12
Take ln of both sides; power rule.
x=ln⁡12ln⁡5x = \dfrac{\ln 12}{\ln 5}
Divide by ln 5.

Practice ★

  1. Solve 6x=356^x = 35. Give the exact answer using natural logarithms.
  2. Solve 2x=102^x = 10. Give the exact answer using natural logarithms.
  3. Solve 7x=367^x = 36. Give the exact answer using natural logarithms.
  4. Solve 4x=204^x = 20. Give the exact answer using natural logarithms.
  5. Solve 2x=172^x = 17. Give the exact answer using natural logarithms.

Watch out!

Practice ★★

  1. Solve 2⋅3x=482\cdot 3^x = 48. Give the exact answer.
  2. Solve 6⋅4x=906\cdot 4^x = 90. Give the exact answer.
  3. Solve 4⋅4x=304\cdot 4^x = 30. Give the exact answer.
  4. Solve 6⋅5x=506\cdot 5^x = 50. Give the exact answer.
  5. Solve 2⋅4x=722\cdot 4^x = 72. Give the exact answer.
  1. Solve 4⋅2x=654\cdot 2^x = 65 step by step.
  2. Solve 3⋅5x=633\cdot 5^x = 63 step by step.

Exit ticket

Three questions to check you have got it.

  1. Solve 2⋅3x=482\cdot 3^x = 48. Give the exact answer.
  2. Solve 6⋅4x=906\cdot 4^x = 90. Give the exact answer.
  3. Solve 4⋅4x=304\cdot 4^x = 30. Give the exact answer.

Summary card

Solving a⋅bx=ca \cdot b^x = c

  1. Isolate the power: bx=cab^x = \dfrac{c}{a}.
  2. Take ln⁡\ln: xln⁡b=ln⁡cax \ln b = \ln \dfrac{c}{a}.
  3. Divide: x=ln⁡(c/a)ln⁡bx = \dfrac{\ln(c/a)}{\ln b}.

Watch out: never multiply aa into the base.

End of the lesson

Next lesson: Solve logarithmic equations and check the domain Back to the chapter

Answers: Solve exponential equations in any base with logarithms

A

  1. ln⁡(35)ln⁡(6)\frac{\ln{\left(35 \right)}}{\ln{\left(6 \right)}}
  2. ln⁡(10)ln⁡(2)\frac{\ln{\left(10 \right)}}{\ln{\left(2 \right)}}
  3. ln⁡(36)ln⁡(7)\frac{\ln{\left(36 \right)}}{\ln{\left(7 \right)}}
  4. ln⁡(20)ln⁡(4)\frac{\ln{\left(20 \right)}}{\ln{\left(4 \right)}}
  5. ln⁡(17)ln⁡(2)\frac{\ln{\left(17 \right)}}{\ln{\left(2 \right)}}

B

  1. ln⁡(24)ln⁡(3)\frac{\ln{\left(24 \right)}}{\ln{\left(3 \right)}}
  2. ln⁡(15)ln⁡(4)\frac{\ln{\left(15 \right)}}{\ln{\left(4 \right)}}
  3. ln⁡(152)ln⁡(4)\frac{\ln{\left(\frac{15}{2} \right)}}{\ln{\left(4 \right)}}
  4. ln⁡(253)ln⁡(5)\frac{\ln{\left(\frac{25}{3} \right)}}{\ln{\left(5 \right)}}
  5. ln⁡(36)ln⁡(4)\frac{\ln{\left(36 \right)}}{\ln{\left(4 \right)}}

C

  1. ln⁡(654)ln⁡(2)\frac{\ln{\left(\frac{65}{4} \right)}}{\ln{\left(2 \right)}}
  2. ln⁡(21)ln⁡(5)\frac{\ln{\left(21 \right)}}{\ln{\left(5 \right)}}

D

  1. ln⁡(24)ln⁡(3)\frac{\ln{\left(24 \right)}}{\ln{\left(3 \right)}}
  2. ln⁡(15)ln⁡(4)\frac{\ln{\left(15 \right)}}{\ln{\left(4 \right)}}
  3. ln⁡(152)ln⁡(4)\frac{\ln{\left(\frac{15}{2} \right)}}{\ln{\left(4 \right)}}

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