Logarithm Calculator

Same argument, other bases

Step-by-step solution

1. 1

   The question

   Find y = log_2(8): the exponent y such that 2 raised to y gives 8.

   $$\log_{2}\left(8\right) = y \iff 2^{\,y} = 8$$
2. 2

   Write both numbers as products of prime powers

   Prime factorizations: 2 = 2 and 8 = 2^3. The exponent vectors are proportional, so the argument is a rational power of the base and the logarithm is exact.

   $$2 = 2, \qquad 8 = 2^{3}$$
3. 3

   Match the exponents

   Both sides are powers of the same prime, so the exponents must be equal.

   $$y = 3$$
4. 4

   Solve for y

   Dividing both sides by the exponent of the base gives y = 3.

   $$y = 3$$
5. 5

   Check

   Raising the base to this power returns the argument: 2^3 = 8.

   $$2^{3} = 8$$
6. 6

   Decimal form

   As a decimal, 3 ≈ 3.

   $$3 \approx 3$$
7. 7

   Result

   log_2(8) = 3 — exact, because 8 is a rational power of 2.

   $$\log_{2}\left(8\right) = 3$$