09.6 - Solving Logarithmic Equations

Answer explanation

To solve the equation \(10^x = 42\), we take the logarithm base 10 of both sides, yielding \(x = \log_{10}(42)\). This is the correct choice, approximately equal to 1.623.

Tags

CCSS.HSF.BF.B.5

Answer explanation

To solve for x in the equation 8^x = 0.00325, we use logarithms. The correct form is x = log_8(0.00325). Calculating this gives approximately -2.755, confirming the first answer choice as correct.

Tags

CCSS.HSF.BF.B.5

• 0.987

• 2.133

• 3.456

• 1.234

Answer explanation

To solve for x, rewrite the equation as 10^{5-3x} = 0.041. Taking the logarithm gives 5 - 3x = log10(0.041). Solving for x yields x = (5 - log10(0.041))/3, which approximates to 2.133.

Tags

CCSS.HSF.LE.A.4

• 5.123

• 7.890

• 2.961

• 6.267

Answer explanation

To solve for x in the equation 3^{x+1}=2007, take the logarithm of both sides: (x+1) log(3) = log(2007). Solving gives x ≈ 2.961, which matches the correct answer choice.

Tags

CCSS.HSF.LE.A.4

• 3.682

• 3.879

• 2.500

• 4.000

Answer explanation

To solve the equation 2·5^x - 29 = 1000, first add 29 to both sides to get 2·5^x = 1029. Then divide by 2, yielding 5^x = 514.5. Taking the logarithm gives x = log(514.5)/log(5) ≈ 3.879, which is the correct answer.

Tags

CCSS.HSF.LE.A.4

• 3.332204

• 2.718281

• 4.000000

• 3.000000

Answer explanation

To solve for x, rearrange the equation: e^x = 40 - 12 = 28. Taking the natural logarithm gives x = ln(28). This evaluates to approximately 3.332204, which is the correct answer.

Tags

CCSS.HSF.LE.A.4

• 3.884

• 5.225

• 4.125

• 6.350

Answer explanation

To solve for x, first isolate e^x: 5e^x = 843, then e^x = 168.6. Taking the natural logarithm gives x = ln(168.6) ≈ 3.884. Thus, the correct answer is 3.884.

Tags

CCSS.HSF.LE.A.4

• x = 5 - ln(2275) ≈ -2.728

• x = 5 + ln(2275) ≈ 12.728

• x = ln(2275) - 5 ≈ 5.728

• x = 5 - ln(2300) ≈ -2.438

Answer explanation

To solve for x, isolate the exponential term: e^{5-x} = 2275. Taking the natural logarithm gives 5 - x = ln(2275). Thus, x = 5 - ln(2275), which approximates to -2.728.

• x ≈ 7.27

• x ≈ 5.13

• x ≈ 8.45

• x ≈ 6.02

Answer explanation

To solve for x, first isolate the exponential term: e^{x+1} = 2(1966 - 9). Then, take the natural logarithm: x + 1 = ln(3934). Finally, solve for x: x ≈ 7.27.

Tags

CCSS.HSF.LE.A.4

• 2.154

• 1.875

• 2.092

• 3.142

Answer explanation

To solve for x in the equation 5^x - 10 = 19, first add 10 to both sides to get 5^x = 29. Then, take the logarithm: x = log(29)/log(5) ≈ 2.092. Thus, the correct answer is 2.092.

Tags

CCSS.HSF.LE.A.4