Free Logarithm Solver
Paste any log equation and get the full working — log rules, change of base, natural log, exponential-to-log conversion. Built for Class 11 CBSE/ICSE, A-Level Maths and JEE Main.
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A worked log equation — product rule, conversion and domain check
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Use the product rule to combine logs
The rule log_b A + log_b B = log_b(AB) lets us merge the two logs on the left.
- 2
Convert from log form to exponential form
By definition, log_b X = Y means X = b^Y. So log_2(...) = 4 means (...) = 2⁴.
- 3
Expand and bring to standard form
Expand the product (x + 3)(x − 1), then subtract 16 from both sides.
- 4
Solve the quadratic using the formula
Apply x = (−b ± √(b² − 4ac)) / 2a with a = 1, b = 2, c = −19.
- 5
Check the domain (this step saves marks)
The original equation requires x + 3 > 0 AND x − 1 > 0 — both log arguments must be positive. So x > 1. Of the two candidates, −1 + 2√5 ≈ 3.47 > 1 ✓ but −1 − 2√5 ≈ −5.47 < 1 ✗. Only the positive root is valid.
Final Answer
How to solve logarithm equations — rules, conversion and domain checks
A logarithm is the inverse operation of exponentiation: if b^y = x then log_b(x) = y. In words, 'log base b of x is the power you need to raise b to in order to get x'. The most common bases are 10 (common log, written log x) and e ≈ 2.718 (natural log, written ln x). Computer science also uses base 2.
The four rules you must memorise are: (1) Product — log_b(AB) = log_b A + log_b B. (2) Quotient — log_b(A/B) = log_b A − log_b B. (3) Power — log_b(A^n) = n · log_b A. (4) Change of base — log_b A = log_c A / log_c B for any new base c (useful when your calculator only does log and ln). The identity log_b b = 1 and log_b 1 = 0 follow from the definition and are worth remembering separately.
Solving a log equation follows a predictable pattern. First, use the log rules to combine multiple logs on the same side into a single log (product rule merges addition, quotient rule merges subtraction). Second, convert from log form to exponential form — this removes the logarithm entirely and usually leaves you with a polynomial. Third, solve the polynomial. Fourth — and this is where most exam marks are lost — check the domain: every log argument in the ORIGINAL equation must be positive, so any root that makes a log argument negative or zero is extraneous and must be discarded.
Exponential equations (where the variable is in the exponent, like 2^x = 32) are the mirror image. You take the log of both sides to bring the variable down from the exponent, then solve the resulting linear or polynomial equation. The solver handles both directions — paste a log equation or an exponential equation and it picks the right method.
Log problems to practise (Class 11, A-Level, JEE)
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.Basic log equationClass 11Easy
- Solve with AI →2.Quotient ruleClass 11Medium
- Solve with AI →
3. 2^{x + 1} = 7
Exponential to logClass 11Medium - Solve with AI →4.Quadratic inside logClass 11 / JEEMedium
- Solve with AI →5.Substitution (u = log_2 x)JEE MainHard
Frequently asked questions
How do I solve a logarithm equation step by step?+
First, use the log rules (product, quotient, power) to combine multiple logs on the same side into a single log. Second, convert from log form to exponential form: log_b X = Y becomes X = b^Y. Third, solve the resulting polynomial. Fourth, check the domain — every log argument in the ORIGINAL equation must be positive, so reject any roots that make an argument ≤ 0.
What are the four logarithm rules?+
Product: log_b(AB) = log_b A + log_b B. Quotient: log_b(A/B) = log_b A − log_b B. Power: log_b(A^n) = n · log_b A. Change of base: log_b A = log_c A / log_c B (useful when your calculator only has log and ln). Memorise these four and you can manipulate almost any log expression.
What is the difference between log and ln?+
log without a subscript usually means log base 10 (common log), and ln means log base e ≈ 2.718 (natural log). Both are logarithms — just with different bases. The natural log is standard in calculus because d/dx [ln x] = 1/x with no extra constant factor; common log appears more in engineering and chemistry.
How do I solve an exponential equation like 3^x = 20?+
Take the log of both sides (either log or ln works). log(3^x) = log 20, which by the power rule becomes x · log 3 = log 20, so x = log 20 / log 3 ≈ 2.727. This is the standard method for any exponential equation where you can't get both sides to the same base.
Why do I need to check the domain after solving a log equation?+
log_b X is only defined when X > 0. When you combine logs using the product or quotient rules, the combined expression's domain may be larger than the original's — so solutions to the combined equation might not be valid for the original. Always substitute each candidate answer into the ORIGINAL equation's log arguments and reject any that give a non-positive value.
What is change of base and when do I need it?+
Change of base lets you convert log_b A into logs of any other base: log_b A = log_c A / log_c B. It's essential when your calculator only has log (base 10) and ln (base e) but your equation uses a different base, like log_2 or log_7. Also used in JEE questions that compare logs with different bases — you convert everything to a common base first.
Does this solver handle logarithmic inequalities?+
Yes, with the warning that logs can flip the direction of an inequality when the base is between 0 and 1 (uncommon in school but it appears in JEE). The solver shows where the base matters and flips the sign explicitly.
How are logarithms used in calculus?+
Two main places: the derivative of ln x is 1/x (the reason natural log exists), and logarithmic differentiation is the standard technique for differentiating complicated products, quotients and powers. The solver handles both — paste a differentiation problem with logs and it applies the right rule.
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