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LogarithmsAlgebraHigh schoolabout 1:23

Solve ln(x^2) = ln(9): animated lesson

A narrated, step-by-step logarithms video lesson with captions. Final answer: x = -3, x = 3.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Solving an equation
    ln⁡(x2)=ln⁡(9)\ln\left({ x}^{2}\right) = \ln\left(9\right)
  2. 2. Step by step
    1. ln⁡(x2)=ln⁡(9)\ln\left({ x}^{2}\right) = \ln\left(9\right)
      Start with the equation
    2. f(x)=ln⁡(x2)−2.1972245773362196=0f(x) = \ln\left({ x}^{2}\right)-2.1972245773362196 = 0
      Rewrite as f(x) = 0
    3. This equation is not polynomial, so solve it numerically
    4. x=−3, x=3x = -3,\ x = 3
      Found 2 solutions in [−10⁹, 10⁹]
  3. 3. The answer
    x = -3, x = 3\text{x = -3, x = 3}

    x = -3, x = 3

  4. 4. Check
    • x = -3: left side = 2.1972246, right side = 2.1972246 ✓
    • x = 3: left side = 2.1972246, right side = 2.1972246 ✓

Transcript

The problem. Here is the problem: ln(x squared) equals ln(9). Solve for x numerically (the equation contains functions such as sin, log or roots).

Step by step. Start with the equation. Rewrite as f(x) equals 0. This equation is not polynomial, so solve it numerically. Scan x from negative 10 to the power 9 to 10 to the power 9 for sign changes of f and for points where the graph touches the axis, then narrow each one down. Every root is checked by substituting it back. Found 2 solutions in [−10 to the power 9, 10 to the power 9].

The answer. So the answer is x equals negative 3, x equals 3. Values marked approximately are numerical approximations accurate to about 10 significant digits. Every value was checked by substitution.

Check. x equals negative 3: left side equals 2.1972246, right side equals 2.1972246, which checks out. x equals 3: left side equals 2.1972246, right side equals 2.1972246, which checks out.

About logarithms

Evaluate logarithms and solve logarithmic equations using the laws of logs.

More lessons like this

Solve ln(x^2) = ln(9): Logarithms Step-by-Step Lesson