Logarithm Properties Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Condense 2ln⁡x−12ln⁡(x+1)+3ln⁡y2\ln x - \frac{1}{2}\ln(x + 1) + 3\ln y into a single logarithm.

Solution

  1. 1
    Apply the power rule: ln⁡x2−ln⁡(x+1)1/2+ln⁡y3\ln x^2 - \ln(x+1)^{1/2} + \ln y^3.
  2. 2
    Apply the product rule to the positive terms: ln⁡(x2y3)−ln⁡x+1\ln(x^2 y^3) - \ln\sqrt{x+1}.
  3. 3
    Apply the quotient rule: ln⁡(x2y3x+1)\ln\left(\frac{x^2 y^3}{\sqrt{x+1}}\right).

Answer

ln⁡(x2y3x+1)\ln\left(\frac{x^2 y^3}{\sqrt{x+1}}\right)
Condensing is the reverse of expanding: use the power rule first to move coefficients into exponents, then combine with the product and quotient rules.

About Logarithm Properties

The three fundamental rules of logarithms: the product rule log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y, the quotient rule log⁡b ⁣(xy)=log⁡bx−log⁡by\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y, and the power rule log⁡b(xn)=nlog⁡bx\log_b(x^n) = n\log_b x.

Learn more about Logarithm Properties →

More Logarithm Properties Examples

Example 1 easy

Expand [formula] using logarithm properties.

Example 3 medium

Use the change of base formula to evaluate [formula] to three decimal places.

Example 4 easy

Expand [formula].

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