Natural Logarithm Math Example 3

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

Example 3

medium
Solve ln⁡(2x+1)=3\ln(2x + 1) = 3 for xx.

Solution

  1. 1
    Exponentiate both sides with base ee: 2x+1=e32x + 1 = e^3.
  2. 2
    Solve: 2x=e3−12x = e^3 - 1, so x=e3−12x = \frac{e^3 - 1}{2}.

Answer

x=e3−12x = \frac{e^3 - 1}{2}
To solve equations with ln⁡\ln, exponentiate both sides using ee as the base. This eliminates the logarithm since eln⁡(a)=ae^{\ln(a)} = a. Always verify the argument of ln⁡\ln is positive: 2x+1=e3>02x+1 = e^3 > 0, which is satisfied.

About Natural Logarithm

The logarithm with base e≈2.71828e \approx 2.71828: ln⁡x=log⁡ex\ln x = \log_e x. It is the inverse function of exe^x.

Learn more about Natural Logarithm →

More Natural Logarithm Examples

Example 1 easy

Evaluate [formula].

Example 2 medium

Simplify [formula].

Example 4 hard

Find the derivative of [formula] and determine where [formula] is increasing.

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