Math Buddyv1.0

Condense logarithms with steps

Type or scan a sum of logarithms. MathBuddy writes it as one logarithm and names each rule it uses.

Loading math input...

What this page covers

  • The power rule, which moves a coefficient into the exponent: kln⁡a=ln⁡akk\ln a = \ln a^k.
  • The product and quotient rules, which turn sums into products and differences into quotients.
  • Constants rewritten as logarithms, such as 2=ln⁡e22 = \ln e^2 or 1=log⁡221 = \log_2 2, so they combine too.
  • Not covered here: the reverse direction (use Expand logarithms calculator) and solving logarithmic equations (use Solve an equation for x).

How to enter the problem

  • Type ln, log or log_2 with each argument in brackets, for example 2ln(3) + 1/2 ln(x) − ln(x + 1).
  • 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

Condense and evaluate log⁡240−log⁡25\displaystyle \log_2 40 - \log_2 5.

Answer

Verified

The answer is

33

Explanation

  1. Apply the quotient rule
    log⁡240−log⁡25=log⁡2405\log_2 40 - \log_2 5 = \log_2 \frac{40}{5}
  2. Simplify the argument
    405=8\frac{40}{5} = 8, so the expression is log⁡28\log_2 8.
  3. Evaluate
    23=82^3 = 8, so log⁡28=3\log_2 8 = 3.

Example 2

Problem

Condense 12ln⁡(x+4)+ln⁡3−2\displaystyle \frac{1}{2}\ln(x + 4) + \ln 3 - 2.

Answer

Verified

The answer is

ln⁡(3x+4e2)\ln\left(\frac{3\sqrt{x+4}}{e^2}\right)

Explanation

  1. Apply the power rule
    12ln⁡(x+4)=ln⁡x+4\frac{1}{2}\ln(x + 4) = \ln\sqrt{x + 4}
  2. Write the constant as a logarithm
    2=ln⁡e22 = \ln e^2
  3. Combine
    ln⁡x+4+ln⁡3−ln⁡e2=ln⁡3x+4e2\ln\sqrt{x + 4} + \ln 3 - \ln e^2 = \ln\frac{3\sqrt{x + 4}}{e^2}

Example 3

Problem

Condense 3log⁡2(x2+1)−12log⁡2x+1\displaystyle 3\log_2(x^2 + 1) - \frac{1}{2}\log_2 x + 1.

Answer

Verified

The answer is

log⁡2(2(x2+1)3x)\log_2\left(\frac{2(x^2+1)^3}{\sqrt{x}}\right)

Explanation

  1. Apply the power rule to both terms
    log⁡2(x2+1)3−log⁡2x\log_2 (x^2 + 1)^3 - \log_2 \sqrt{x}.
  2. Write 1 in base 2
    1=log⁡221 = \log_2 2
  3. Combine
    The added logs multiply and the subtracted one divides: log⁡22(x2+1)3x\log_2\frac{2(x^2 + 1)^3}{\sqrt{x}}.

Common mistakes

  • Turning a sum of logs into the log of a sum: ln⁡(x2+1)+ln⁡4=ln⁡(4x2+4)\ln(x^2 + 1) + \ln 4 = \ln(4x^2 + 4), not ln⁡(x2+5)\ln(x^2 + 5).
  • Combining before applying the power rule. The coefficient goes into the exponent first: 2ln⁡3=ln⁡92\ln 3 = \ln 9.
  • Putting a subtracted logarithm in the numerator. A minus sign sends its argument to the denominator.
  • Combining logs with different bases. log⁡2x+ln⁡x\log_2 x + \ln x cannot become one logarithm without a change of base.

Checks, assumptions and limits

  • Expand the single logarithm with the same rules in reverse; the original expression should come back term by term.
  • Compare values at one point. At x=4x = 4, 2ln⁡3+12ln⁡4−ln⁡5≈1.2812\ln 3 + \frac{1}{2}\ln 4 - \ln 5 \approx 1.281, and ln⁡9⋅25=ln⁡3.6≈1.281\ln\frac{9 \cdot 2}{5} = \ln 3.6 \approx 1.281.
  • The log rules assume positive arguments, so the condensed form matches the original where x>0x > 0.
  • MathBuddy can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

The power rule, the product rule and the quotient rule. Each step is titled with the rule it applies, so you can follow the order.

Yes, as long as every term has the same base. Constants are rewritten in that base before they are combined.

It is good practice. The rules assume positive arguments, so the condensed form equals the original only where every argument is positive. Ask a follow-up if you want that worked out.

1 math question to try today without signing in, 3 a day signed in, unlimited with Pro.

Ready to solve your next math problem?

Try one problem free in your browser. Create a free account for 3 math questions a day.

Use the web solverGet the Android appGet the iPhone app

Last updated: · MathBuddy