Math Buddyv1.0

Inverse natural log calculator

Type an equation with ln or log. MathBuddy undoes the logarithm and solves for x.

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What this page covers

  • The inverse of the natural log: if ln⁡x=k\ln x = k, then x=ekx = e^k.
  • The inverse of the common log: if log⁡x=k\log x = k, then x=10kx = 10^k.
  • Equations where the logarithm holds an expression, such as ln⁡(2x−1)=4\ln(2x - 1) = 4, with exact and decimal answers.
  • Not covered here: logarithms in other bases (use the log base 2 calculator or the logarithm calculator), and equations with several logarithms, which you can still type and try.

How to enter the problem

  • Type it or paste it. Write ln for the natural log and log for base 10, for example ln(2x - 1) = 4.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Solve log⁡x=−2\displaystyle \log x = -2, where log⁡\displaystyle \log means base 10.

Answer

Verified

The answer is

x=1100x = \frac{1}{100}

Explanation

  1. Undo the logarithm
    The inverse of base-10 log is the power of 10, so x=10−2x = 10^{-2}.
  2. Write it as a fraction
    10−2=110010^{-2} = \frac{1}{100}

Example 2

Problem

Solve ln⁡(2x−1)=4\displaystyle \ln(2x - 1) = 4.

Answer

Verified

The answer is

x=e4+12x = \frac{e^{4} + 1}{2}

Explanation

  1. Undo the natural log
    Raise ee to both sides
    2x−1=e42x - 1 = e^4
  2. Solve the linear equation
    2x=e4+12x = e^4 + 1, so x=e4+12x = \frac{e^4 + 1}{2}.
  3. Check the domain
    2x−1=e42x - 1 = e^4 is positive, so the logarithm is defined. As a decimal, x≈27.80x \approx 27.80.

Example 3

Problem

Find x\displaystyle x if ln⁡x=1.5\displaystyle \ln x = 1.5, to 4 decimal places.

Answer

Verified

The answer is

x=e1.5≈4.4817x = e^{1.5} \approx 4.4817

Explanation

  1. Undo the natural log
    x=e1.5x = e^{1.5}
  2. Evaluate
    e1.5=4.48168…e^{1.5} = 4.48168\ldots, which rounds to 4.48174.4817.

Common mistakes

  • Multiplying instead of raising to a power: ln⁡x=3\ln x = 3 gives x=e3x = e^3, not x=3ex = 3e.
  • Taking 1÷ln⁡31 \div \ln 3 as the inverse. The inverse of the logarithm is the exponential, not the reciprocal.
  • Using ee when the equation uses base-10 log. log⁡x=−2\log x = -2 gives 10−210^{-2}, not e−2e^{-2}.

Checks, assumptions and limits

  • Check by hand: put the answer back in. ln⁡(e3)=3\ln(e^3) = 3, so x=e3x = e^3 fits.
  • A logarithm is defined only for positive inputs. Any answer that makes the inside of the log zero or negative is rejected.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

The exponential function. If ln x equals a number k, then x equals e to the power k.

Yes. Ask for a decimal, or say how many decimal places you want, and the answer is rounded at the last step.

Yes. Write log for base 10. The inverse is then 10 to the power of the right side.

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