A logarithm answers one question: what exponent turns the base into the argument? log₂ 32 asks "2 to what power is 32?" and the answer is 5.
Rewriting the log in exponential form makes every problem in this DeltaMath skill mechanical, including fractions, decimals, and the special bases 10 and e.
How to solve it, step by step
Rewrite as an exponential equation
log_b(a) = x means b^x = a. For log₃ 81: 3^x = 81.
Express the argument as a power of the base
81 = 3⁴, so x = 4.
Use negative exponents for fractions
log₅ (1/25): 5^x = 1/25 = 5⁻², so x = −2.
Remember the special bases
log with no base is base 10: log 1000 = 3, log 0.001 = −3. ln is base e: ln e⁴ = 4, ln 1 = 0.
Worked example
Example 1
Evaluate log₄ 8.
- 4^x = 8
- Write both as powers of 2: (2²)^x = 2³
- 2^(2x) = 2³ → 2x = 3
- x = 3/2
Answer 3/2
Example 2
Evaluate log₉ (1/3).
- 9^x = 1/3
- (3²)^x = 3⁻¹ → 2x = −1
Answer −1/2
Mistakes DeltaMath marks wrong
- Answering with the argument's exponent in a different base (writing 3 for log₄ 8 because 8 = 2³).
- Confusing log_b(1) = 0 with log_b(b) = 1.
- Treating log 100 as base e or ln 100 as base 10.
- Trying to evaluate log₂ (−8); logarithms of negative numbers are undefined for real numbers.
How to type the answer
- Enter integers or fractions: 4, -2, 3/2.
- For undefined cases pick "undefined" if offered.