Worksheet: Solving Exponential Equations Using Logarithms

In this worksheet, we will practice using logarithms to solve exponential equations.

Q1:

Find, to the nearest hundredth, the value of ๐‘ฅ for which 2=9๏—๏Šฐ๏Šฎ.

Q2:

Use a calculator to find the value of ๐‘ฅ for which 9=8๏—. Give your answer correct to two decimal places.

Q3:

Given that 2=18๏— and ๐‘›<๐‘ฅ<๐‘›+1, where ๐‘› is an integer, determine the value of ๐‘›.

Q4:

Find the value of ๐‘ฅ for which 8=12.7๏Šฉ๏Šฑ๏Šจ๏—. Give your answer to the nearest tenth.

Q5:

Use a calculator to find the value of ๐‘ฅ for which 3=8๏Šฑ๏Šช๏—๏Šฑ๏Šฉ๏—๏Šฐ๏Šช๏Ž–๏Šญ. Give your answer correct to two decimal places.

Q6:

Find the solution set to the equation 15=64๏—. Give your answer to three decimal places.

  • A{1.536}
  • B{0.651}
  • C{0.234}
  • D{4.267}

Q7:

Use a calculator to find the value of ๐‘ฅ for which 2รท5=2๏Šจ๏Šฑ๏Šซ๏—๏—๏Šฐ๏Šฉ. Give your answer correct to two decimal places.

Q8:

Use a calculator to find the value of ๐‘ฅ for which ๐‘ฅ=97.4๏Šญ๏Ž–๏Šง. Give your answer correct to two decimal places.

Q9:

Find the value of ๐‘ฅ for which 36=2๏Šฉ๏—. Give your answer to the nearest hundredth.

Q10:

Solve 2โ‹…3=5โ‹…4๏—๏— for ๐‘ฅ, giving your answer to three decimal places.

  • A๐‘ฅ=3.793
  • B๐‘ฅ=3.185
  • C๐‘ฅ=0.369
  • D๐‘ฅ=โˆ’0.369
  • E๐‘ฅ=โˆ’3.185

Q11:

Solve 3=11๏— for ๐‘ฅ, giving your answer to three decimal places.

  • A๐‘ฅ=0.497
  • B๐‘ฅ=2.183
  • C๐‘ฅ=3.667
  • D๐‘ฅ=0.564
  • E๐‘ฅ=0.273

Q12:

Estimate the solution of 16=9๏— to one decimal place, given that log๏Šง๏Šฆ2โ‰ˆ0.3 and log๏Šง๏Šฆ6โ‰ˆ0.8.

Q13:

Solve 17=2๏— for ๐‘ฅ, giving your answer to three decimal places.

  • A๐‘ฅ=0.118
  • B๐‘ฅ=0.370
  • C๐‘ฅ=0.929
  • D๐‘ฅ=3.214
  • E๐‘ฅ=0.245

Q14:

Solve 20=50(1.029)๏— for ๐‘ฅ, giving your answer to three decimal places.

  • A๐‘ฅ=32.052
  • B๐‘ฅ=โˆ’32.052
  • C๐‘ฅ=0.389
  • D๐‘ฅ=61.680
  • E๐‘ฅ=โˆ’0.389

Q15:

Solve 12(0.85)=48๏—. Give your answer to two decimal places.

  • A๐‘ฅ=โˆ’8.53
  • B๐‘ฅ=โˆ’0.12
  • C๐‘ฅ=โˆ’15.29
  • D๐‘ฅ=โˆ’23.82

Q16:

Use a calculator to find the value of ๐‘ฅ for which 2ร—4=32ร—3๏—๏Šฑ๏Šซ๏Šจ๏—๏Šฐ๏Šฉ. Give your answer correct to two decimal places.

Q17:

Use a calculator to find the value of ๐‘ฅ for which 2ร—7=16ร—7๏—๏—๏Šฐ๏Šฏ. Give your answer correct to two decimal places.

Q18:

Use a calculator to find the value of ๐‘ฅ for which 5+5=208๏—๏Šฐ๏Šง๏—๏Šฑ๏Šง. Give your answer correct to two decimal places.

Q19:

Find the solution set of ๐‘ฅ=81log๏Žข๏—.

  • A๏ฌ9,19๏ธ
  • B๏ฌ6,16๏ธ
  • C๏ฌ3,13๏ธ
  • D{2,โˆ’2}
  • E๏ฌ81,181๏ธ

Q20:

Use a calculator to find the value of ๐‘ฅ for which ๏€ผ23๏ˆ=1.7602๏—. Give your answer correct to two decimal places.

Q21:

Solve for ๐‘ฅ2=3:๏—๏Šฐ๏Šฉ๏Šจ๏—๏Šฑ๏Šง.

  • A๐‘ฅ=33โˆ’22โˆ’2โˆ’33loglogloglog
  • B๐‘ฅ=3+3223โˆ’2loglogloglog
  • C๐‘ฅ=23โˆ’23โˆ’32loglogloglog
  • D๐‘ฅ=2+233โˆ’22loglogloglog

Q22:

Solve for ๐‘ฅ2=12:๏—๏Šฐ๏Šซ.

  • A๐‘ฅ=212โˆ’5loglog
  • B๐‘ฅ=16โˆ’5log
  • C๐‘ฅ=6โˆ’5log
  • D๐‘ฅ=122โˆ’5loglog

Q23:

Given that 6=6ร—3๏—๏Šฑ๏Šญ๏—๏Šฑ๏Šญ, find the value of ๐‘ฅ correct to the nearest hundredth.

Q24:

Given that 3๐‘ฅโˆ’4=1๏Žข๏Žก, find the value of ๐‘ฅ. If necessary, give your answer to 3 decimal places.

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