What is the answer of the logarithmic equation?

Question

Solve the logarithmic equation. Give the answer to two significant figures.

3lnx+8=ln(3x)+12.18

Answer 1

3lnx + 8 = ln(3x) + 12.18

 

ln(x³) - ln(3x) = 4.18

 

ln[x³/(3x)] = 4.18

 

ln[x²/3] = 4.18

 

x²/3 = e^4.18

 

x² = 3e^4.18       So, x = ±√(3e^4.18)

 

x can't be negative since otherwise ln(3x) would be undefined.  

 

x = √(3e^4.18) ≈ 14

Answer 2

3 ln x + 8 = ln (3x) + 12.18

 

Use the rule about the logarithm of a product to simplify ln (3x):

3 ln x + 8 = ln (3) + ln x + 12.18

 

Subtract ln x from both sides:

2 ln x + 8 = ln (3) + 12.18

 

Subtract 8 from both sides:

2 ln x = ln (3) + 4.18

 

Divide both sides by 2:

ln x = ln (3) * (1/2) + 2.09

 

Exponentiate both sides:

e^{ln x} = e ^{ln (3) * (1/2) + 2.09}

 

Follow the rules about exponentiating a logarithm:

x = 3^1/2 * e^2.09

x = √3 * e^2.09

x = 14.00

 

If they want it to two significant figures, the answer should be stated as 14.  If they want it to two decimal places, the answer should be stated as 14.00.